%% Eduardo Martinez Pedroza.
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\begin{document}

\title{Maass space for lifting to GL(2,B) over a division quaternion algebra}
\author{Siddhesh V. Wagh}
\date{May 2019}

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UNIVERSITY OF OKLAHOMA
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GRADUATE COLLEGE
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MAASS SPACE FOR LIFTING TO GL(2,B)
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OVER A DIVISION QUATERNION ALGEBRA
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A DISSERTATION
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SUBMITTED TO THE GRADUATE FACULTY
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in partial fulfillment of the requirements for the
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Degree of
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DOCTOR OF PHILOSOPHY
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By
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Siddhesh Wagh
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Norman, Oklahoma
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2019
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MAASS SPACE FOR LIFTING TO GL(2,B)
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OVER A DIVISION QUATERNION ALGEBRA
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A DISSERTATION APPROVED FOR THE
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DEPARTMENT OF MATHEMATICS
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BY
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\  \  \  \  \  \  \  & Dr. Ameya Pitale, Chair\\
 \\ &  \\
  \\ &  \\
  & Dr. Ralf Schmidt \\
\\ & \\
 \\ &  \\
 & Dr. Tomasz Przebinda \\
\\   & \\
 \\ &  \\
 & Dr. Alan Roche \\
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  \\ &  \\
   & Dr. Ramkumar Parthasarathy\\
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\copyright \    Copyright by Siddhesh Wagh \   2019

All Rights Reserved.
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\chapter*{Acknowledgements}

First, I wish to express my gratitude to my research advisor, Professor Ameya Pitale, for teaching me a great deal of mathematics,
for encouraging to explore and work in mathematics,  for his guidance and unconditional support during all these years of graduate school,
and for being a excellent mentor.  I am grateful for having had the opportunity to work with him.

I also wish to thank my parents Shubhada and Vivek Wagh for encouraging me to pursue math, my sister Shreeya Wagh for being a companion in all my foolish endeavors and my girlfriend Supriya Kelkar for loving me despite of all my oddities. I would also like to the friends Ihave made over the years from OU, from CMI and from my childhood, who believed in me and gave me the strength to persevere during my moments of weakness. 

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\chapter{Introduction}

One of the fundamental problems in the theory of automorphic forms or
representations is the Ramanujan conjecture. Originally formulated by Ramanujan as estimation for the Fourier coefficients of the weight 12 holomorphic cusp form $\Delta$ over $\SL_2(\Z)$ on the upper half plane $\mathfrak{h}$, the conjecture has been generalized to functions over a broader set of groups in terms of local representations of the associated automorphic forms. To review it, let $\mathcal{G}$ be a reductive algebraic group over a number field $F$, and let $\A := \otimes_{\nu \leq \infty} 'F_\nu$ be the ring of adeles for $F$, where $F_\nu$ denotes the local field at a place $\nu$. Then, one of the old versions of the Ramanujan conjecture can be stated as follows :

\begin{namedthm}\textbf{Conjecture:}
	Let $\pi \simeq \otimes_{\nu \leq \infty}' \pi_\nu$ be an irreducible cuspidal representation of $\mathcal{G}(\A)$, where $\pi_\nu$ denotes the local component of $\pi$ at the place $\nu$. Then $\pi_\nu$ is tempered for every $\nu \leq \infty$.
\end{namedthm}

This naive version of the Ramanujan conjecture is known to be false with the first numerical counter examples being found by Saito and Kurokawa \cite{Kuro}. Adrianov \cite{And1}, Maass \cite{Maass} and Zagier \cite{Zag} showed that the Saito-Kurokawa lift from elliptic cusp forms to holomorphic Siegel cusp forms of degree two always violates the conjecture. Maass found explicit relations between the Fourier coefficients of the holomorphic Siegel cusp forms which characterize the image of the lift (cf. \cite{Maass}). We shall refer to these as the Maass relations and to the image as the Maass space. In \cite{16}, Ikeda generalized the process of Saito-Kurokawa lifts for holomorphic Siegel cusp forms of higher degree. Kohnen and Kojima characterize the Maass space for Ikeda lifts again via a similar process as that of Maass (cf. \cite{17}, \cite{18}). Both these proofs rely crucially on intermediate spaces of Jacobi forms.

While this naive version of the Ramanujan conjecture is strongly believed for the general linear groups, the generalized version is expected only for generic cuspidal representations of quasisplit reductive groups. 
Muto, Narita and Pitale in \cite{MNP} provide a counterexample to the Ramanujan conjecture for $\GL_2(B)$ over the division quaternion
algebra $B$ with discriminant two. Note that $\GL_2(B)$ is an inner form of the
split group $\GL_4$. 
Unlike the cases of Saito-Kurokawa and Ikeda, the construction in \cite{MNP} does not involve any intermediate spaces of Jacobi forms. 
Instead, given Fourier coefficients $c(N)$ of $f \in \SG$ which is an eigenfunction of the Atkin-Lehner involution, they directly define numbers $A(\beta)$ (cf. \eqref{3.3}). 
Then they show that these $A(\beta)$ are the Fourier coefficients of some $F_f \in \MG$ by using Maass Converse Theorem (cf. Theorem \ref{Pitale}). 
Here $\MG$ is the space of Maass forms on the 5-dimensional hyperbolic space with respect to $\GL_2(\mathcal{O})$, where $\mathcal{O}$ is the Hurwitz order in $B$ (see Section 2.2 for details). 
They further show that if $f$ is a Hecke eigenform, then so is $F_f$ and the representation $\Pi_F \simeq \otimes'_{p \leq \infty} \Pi_{F,p}$ of $\GL_2(B_\A)$ corresponding to $F_f$ is a counterexample to the Ramanujan conjecture. 
They also show that the image of $\Pi_F$ under the global Jacquet-Langlands correspondence is the irreducible constituent of $\ind_{P_{2,2}(\A)}^{\GL_4(\A)} (|\det|_\A^{-1/2} \sigma_f \times |\det|_\A^{1/2} \sigma_f)$, where $\sigma_f$ is the automorphic representation of $\GL_2(\A)$ corresponding to $f$.

The question we want to answer here is the same as the one Maass answered for the Saito-Kurokawa case in \cite{Maass}. More precisely, we want to characterize the image of this lift, possibly in terms of recurrence relations between their Fourier coefficients. We tackle this problem by first noticing that $A(\beta)$ depends only on $K = |\beta|^2,u$ and $n$ when $\beta = \varpi_2^un\beta_0$ as in \eqref{3.2} (cf. \ref{rec}). 
\begin{dfn} 
	Let $\Ms$ denote the subspace of \\$\MG$ consisting of functions $F$ whose Fourier coefficients\\ $A(\beta)$ satisfy:
	\begin{enumerate}
		\item If $\beta = \varpi_2^u n\beta_0 $ as in (\ref{3.2}), then $A(\beta)$ depend only on $K = |\beta|^2$,\\ $u$  and $n$. We shall then write $A(\beta)$ as $A(K,u,n)$.
		\item $A(K,u,n)$ satisfy the recurrence relations : 
		\begin{enumerate}
			\item $A(K,u,n) = \frac{-3\epsilon}{\sqrt{2}}A(\frac{K}{2},u-1,n) - A(\frac{K}{4},u-2,n)$ for some $\epsilon \in \{\pm 1\}$, 
			\item  $A(K,u,n)= \sum_{d \mid n} d \cdot A(\frac{K}{d^2},u,1)$ .
		\end{enumerate}	
	\end{enumerate}
\end{dfn}
Note that there are no intermediate spaces of Jacobi forms. As a result, we cannot just generalize any of the previous proofs of Maass, Kohnen or Kojima to this case. Instead we take a completely different approach. 

We would like to mention that there is a recent work of Pitale-Saha-Schmidt (c.f. \cite{PSS}) which provides a representation theoretic approach to Saito-Kurokawa lifts which does not use Fourier-Jacobi forms. We note however that the paper is only able to show a one way implication without Jacobi forms, where as we wish to prove both ways.

It is easy to see that the Fourier coefficients of $F_f$ satisfy condition (1). To show that $A(K,u,n)$ also satisfy condition $\eqref{2a}$ and $\eqref{2b}$, we use Legendre's three-square theorem to isolate $c(N)$ as follows:
\begin{prop} 
	Let $N = 4^a b$, where $a,b$ are a non-negative integers and $4 \nmid b $. With assumptions as in Theorem \ref{Th1}, we get
	\begin{equation}
	c(-N) = \frac{A(2N,u,1)}{\sqrt{2N}} + \epsilon \frac{A(N,u-1,1)}{\sqrt{N}}
	\end{equation}
	where \[ 
	u = 
	\begin{dcases}
	2a & \text{if }b \equiv 1,3 \mod (4), \\
	2a+1 & \text{if }b \equiv 2 \mod (4). \\ 
	\end{dcases}
	\].
\end{prop}
Now, we manipulate the defining sum of $A(\beta)$ (c.f. \eqref{3.3}) using these $c(N)$ to show that $A(K,u,n)$ indeed satisfy the recurrence relation \eqref{2b}. The relation \eqref{2a} follows from the fact that $F_f$ is a Hecke eigenform at $p=2$. Hence, we get
\begin{thm}
	Let $f \in \SG$ be an Atkin-Lehner eigenform with eigenvalue $\epsilon \in \{\pm 1 \}$ and which is a Hecke eigenform at $p=2$. Then $\F$ obtained in Theorem \ref{Pitale} belongs to the Maass space $\M^*(\GL_2(\mathcal{O}),r)$.
\end{thm}

This allows us to determine a ``necessary" condition for any $F \in \MG$ to be a lift. We would like to show a theorem that this is also a ``sufficient" condition. If $F \in \Ms$, we can still isolate $c(N)$ as before and now the question reduces to showing these are the Fourier coefficients of some $f \in \SG$.
As a first approach one can try to use the Maass converse theorem to show the automorphy of a function $f$ with Fourier coefficients $\{c(N)\}$. The difficulty is that the analytic properties of the Dirichlet series associated with $F$ do not translate into analytic properties of Dirichlet series obtained from $\{c(N)\}$. To approach this problem by representation theory, we first add the condition that $F$ is a Hecke eigenform for all primes $p$ and obtain the following theorem.

\begin{thm}\label{Th1.3}
	Let $F \in \mathcal{M}^*(\GL_2(\mathcal{O},r)$ such that $F$ is a cuspidal Hecke eigenform. Then, there exists $ f \in \SG$, a Hecke eigenform, such that $F = F_f$.
\end{thm}
We denote by $\Pi_F \simeq \otimes_{p \leq \infty} \Pi_{F,p}$ the automorphic representation of $\GL_2(B_\A)$ associated with $F$. Let the image of $\Pi_F$ under the global Jacquet Langlands map be $\Pi \simeq \otimes_{p \leq \infty} \Pi_p$, a representation of $\GL_4(\A)$. For a cuspidal representation $\sigma$ of $\GL_2(\A)$, we denote by $\MW(\sigma,2)$ the Langlands quotient of $\ind_{P_{2,2}(\A)}^{\GL_4(\A)} (|\det|_\A^{1/2} \sigma \times |\det|_\A^{-1/2} \sigma)$, following the notation of Badulescu and Renard from \cite{4}. The strategy of the proof now is to show that $\Pi = \MW(\sigma,2)$ for $\sigma$ an irreducible cuspidal automorphic representation of $\GL_2(\A)$. We show that $\Pi_{F,p}$ is the unique irreducible constituent of some unramified principal series representation $\ind_{\mathcal{B}_4(\Q_p)}^{\GL_4(\Q_p)}(\chi_1 \times \chi_2 \times \chi_3 \times \chi_4)$ where each $\chi_i$ is an unramified character of $\Q_p^\times$ described in the following proposition. 
\begin{prop}
	For every odd prime $p$, there is a $ \lambda_{p} \in \C$ such that, up to the action of the Weyl group, $\chi_i$ are given by the formula
	\begin{align*} 
	\chi_1(p) = p^{1/2}\frac{\lambda_p + \sqrt{\lambda_p ^2 - 4}}{2} &; \quad \chi_2(p) = p^{1/2}\frac{\lambda_p - \sqrt{\lambda_p ^2 - 4}}{2};\\
	\chi_3(p) = p^{-1/2}\frac{\lambda_p + \sqrt{\lambda_p ^2 - 4}}{2} &; \quad \chi_4(p) = p^{-1/2}\frac{\lambda_p - \sqrt{\lambda_p ^2 - 4}}{2} \numberthis
	\end{align*}
\end{prop}
This is the most crucial result of the paper. The fact that $\chi_i$ are related in this special way and are not arbitrary is an important consequence of the action of the Hecke algebra and recurrence relations from Definition 1.1 \eqref{2b}.
For $p=2$, the structure of the local component $\Pi_{F,2}$ can be obtained from the action of the Hecke algebra and relation \eqref{2a}. The component $\Pi_{F,\infty}$ follows from Section 6.1 of \cite{MNP}. Conditions on the Satake parameters give us that $\Pi$ is indeed of the form $\MW(\sigma,2)$ for some $\sigma$ representation of $\GL_2(\A)$. For an odd prime $p$, let $\chi_{p}$ be the unramified character of $\Q_p^\times$ such that $\chi_{p} (p) = \frac{\lambda_p + \sqrt{\lambda_p ^2 - 4}}{2}$. At the prime $p= \infty$, let $\chi_{\infty}(a) = |a|^s$ where $s = \frac{\sqrt{-1}r}{2}$. For the prime $p=2$, let $\chi$ be an unramified character of $\Q_2^\times$ with $\chi(2) = - \epsilon$ for $\epsilon$ as in condition \eqref{2a} of Definition \ref{rec}.
\begin{prop}
	Let $\sigma = \otimes_{p \leq \infty} \sigma_p$ be the irreducible cuspidal automorphic representation of $\GL_2(\A)$ such that $\Pi = \MW(\sigma,2)$ as above. Then
	\begin{equation}
	\sigma_p = 
	\begin{cases}
	\ind_{\mathcal{B}_2(\Q_p)}^{\GL_2(\Q_p)}(\chi_{p}\times \chi_{p}^{-1}) \quad &\text{ for odd } p< \infty, \\
	\chi St_{\GL_2} \quad &\text{ for } p = 2, \\
	\ind_{\Bg(\R)}^{\GL_2(\R)}(\chi_\infty \times \chi_\infty^{-1}) \quad &\text{ for } p = \infty.
	\end{cases}
	\end{equation}
\end{prop} 
We then look at the distinguished vector in $\sigma$ to find a function $f$ associated to $\sigma$. As $\sigma_2$ is Steinberg and $\sigma_\infty$ is principal series, we can show that $f \in \SG$ as required. We complete the proof by showing that $c(N)$ are indeed the Fourier coefficients of $f$ implying $F =F_f$. 

To generalize Theorem \ref{Th1.3} to all $F \in \Ms$ we first show that $\MG$ is finite dimensional and has a Hecke eigenbasis of operators that commute with their adjoint. With this, proving that $\Ms$ is a Hecke invariant subspace suffices as this implies that $\Ms$ has a Hecke eigenbasis $F_i \in \Ms$ which are lifts of $f_i \in \SG$. By linearity of the defining condition $\eqref{3.3}$, $F = \sum_i a_i F_i$ would be a lift of $\sum_i a_i f_i$. 

We prove that $\Ms$ is Hecke invariant by showing that for all the Hecke operators $T_i$, the image under their action $T_i(F)$ satisfies the conditions of Definition \ref{rec}. The condition that Fourier coefficients of $T_i(F)$ depend only on $K,u$ and $n$ is obtained by writing the coefficients of $T_i(F)$ in terms of $A(K,u,n)$ the Fourier coefficients of $F$. Since each of these coefficients depends only on $K,u$ and $n$, so do the coefficients of $T_i(F)$. Condition \eqref{2a} is equivalent to $F$ being a Hecke eigenform at prime $p=2$ so it is valid for all $F \in \Ms$. Condition \eqref{2b} is shown by computing the recurrence sum for  $(T_{i,p}F)(K,u,n)$ and showing that it is equal to $\sum_{d \mid n}d (T_{i,p}F)(K/d^2,u,1)$. Hence we get the result:

\begin{thm} 
	The following are equivalent.
	\begin{enumerate}
		\item $F$ is a lift from an Atkin-Lehner eigenform $f \in \SG$ with eigenvalue $\epsilon \in \{\pm 1 \}$ and which is a Hecke eigenform at $p=2$.
		\item $F$ is an element of the space $\Ms$
	\end{enumerate}
	%Let $f \in \SG$ be an Atkin-Lehner eigenform with eigenvalue $\epsilon \in \{\pm 1 \}$ and which is a Hecke eigenform at $p=2$. Then $\F$ with Fourier coefficients $A(\beta)$ belongs to the Maass space $\Ms$ and for every $F \in \Ms$ there exists a $f$ such that $F = F_f$. 
\end{thm}


\chapter{Automorphic forms on 5-dimensional hyperbolic space}

\section{Algebraic groups and \\the 5-dimensional hyperbolic space}
Following the notation of Muto, Narita and Pitale in \cite{MNP}, let $B$ be the definite quaternion algebra over $\mathbb{Q}$ with discriminant 2. In terms of the basis $\{1,i,j,k\}$, $B = \Q +\Q i +\Q j +\Q k$ with $i,j,k$ satisfying $$i^2 =j^2=k^2 =-1, ij = -ji=k.$$
$\GL_2(B)$ will be the group of elements of $M_2(B)$ whose reduced norms are non-zero. 
Let $\mathbb{H}= B \otimes_\Q \R$ be the Hamilton quaternion algebra with $x \rightarrow \bar{x}$ the main involution of $\mathbb{H}$. 
Let $tr(x) = x + \bar{x}$ and $\nu(x) = x \bar{x}$ be the reduced trace and reduced norm of $x \in \mathbb{H}$ respectively, with $|x| = \sqrt{\nu(x)}$.

The general linear group $G \coloneqq \GL_2(\mathbb{H})$  admits an Iwasawa decomposition 
$$G =Z^+ NAK,$$
where
\begin{align*}
&Z^+ \coloneqq \left\{\begin{bmatrix}
c & 0 \\
0 & c \\
\end{bmatrix}\Bigg|c \in \R^\times_+\right\}, 
N \coloneqq \left\{n(x) = \begin{bmatrix}
1 & x \\
0 & 1 \\
\end{bmatrix}\Bigg|x\in \mathbb{H}\right\}, \\
& A \coloneqq \left\{a_y = \begin{bmatrix}
\sqrt{y} & 0 \\
0 & \sqrt{y}^{-1} \\
\end{bmatrix} \Bigg|y\in \mathbb{R}^+\right\},
K \coloneqq \left\{k \in G: \prescript{t}{}{}\bar{k}k = 1_2 \right\}.
\end{align*}
The quotient $G/Z^+K$ can be realized as 
$$\left\{\begin{bmatrix}
y & x \\
0 & 1 \\
\end{bmatrix}\Bigg|y\in \R^\times_+, x \in \mathbb{H}\right\}.$$
This gives a realization of the 5-dimensional real hyperbolic space.

\section{Lie algebras}
The Lie algebra $\mathfrak{g}$ of $G$ is $M_2(\HH)$ and has an Iwasawa decomposition 
$$\mathfrak{g} = \mathfrak{z} \oplus \mathfrak{n} \oplus \mathfrak{a} \oplus \mathfrak{k}$$.
Here 
\begin{align*}
&\mathfrak{z} \coloneqq \left\{\begin{bmatrix}
c & 0 \\
0 & c \\
\end{bmatrix}\Bigg|c \in \R \right\}, 
\mathfrak{n} \coloneqq \left\{\begin{bmatrix}
0 & x \\
0 & 0 \\
\end{bmatrix}\Bigg|x\in \mathbb{H}\right\}, \\
& \mathfrak{a} \coloneqq \left\{\begin{bmatrix}
t & 0 \\
0 & -t \\
\end{bmatrix} \Bigg|t \in \mathbb{R} \right\},
\mathfrak{k} \coloneqq \left\{X \in M_2(\HH)| \prescript{t}{}{}\bar{X}+ X = 0_2 \right\}.
\end{align*}
where $\mathfrak{z}, \mathfrak{n}, \mathfrak{a}$ and $\mathfrak{k}$ are the Lie algebras of $Z^+, N, A$ and $K$ respectively.

To consider the root space decomposition of $\mathfrak{g}$ with respect to $\mathfrak{a}$, let $H \coloneqq \begin{bmatrix}
1 & 0 \\
0 & -1 \\
\end{bmatrix}$ and let $\alpha$ be the linear form of $\mathfrak{a}$ such that $\alpha(H)= 1$. Then $\{\pm2\alpha\}$ is the set of roots for $(\mathfrak{g},\mathfrak{a})$. For $z \in \HH$ we put 
$$E^{(z)}_{2\alpha} \coloneqq \begin{bmatrix}
0&z\\
0&0\\
\end{bmatrix}, 
E^{(z)}_{-2\alpha} \coloneqq \begin{bmatrix}
0&0\\
z&0\\
\end{bmatrix}.$$
The set $\{E^{(1)}_{2\alpha},E^{(i)}_{2\alpha},E^{(j)}_{2\alpha},E^{(k)}_{2\alpha}\}$ (respectively $\{E^{(1)}_{-2\alpha},E^{(i)}_{-2\alpha},E^{(j)}_{-2\alpha},E^{(k)}_{-2\alpha}\}$)
form a basis of $\mathfrak{n}$ (respectively a basis of $\bar{\mathfrak{n}} \coloneqq \left\{\begin{bmatrix}
0 & 0 \\
x & 0 \\
\end{bmatrix}\Bigg|x\in \mathbb{H}\right\}$). 
Let $\mathfrak{z}_\mathfrak{a}(\mathfrak{k}) \coloneqq \{ X \in \mathfrak{k} \mid [X,A] = 0 \hspace{0.1 in} \forall A \in \mathfrak{a}\}$, then $\mathfrak{g}$ decomposes into 
$$\mathfrak{g} = (\mathfrak{z} \oplus \mathfrak{z}_\mathfrak{a}(\mathfrak{k}) \oplus\mathfrak{a}) \oplus \mathfrak{n} \oplus \bar{\mathfrak{n}}.$$
Consider the Lie group $\SL_2(\HH)$ consisting of elements in $\GL_2(\HH)$ with their reduced norms 1. Its Lie algebra is $\mathfrak{g}_0 = \mathfrak{s}\mathfrak{l}_2(\HH)$ consisting of elements in $M_2(\HH)$ with their reduced traces zero. 

We introduce the differential operator $\Omega$ defined by the infinitesimal action of 
\begin{equation} \label{Casimir}
\Omega \coloneqq \frac{1}{32}H^2 - \frac{1}{4}H + \frac{1}{8} \sum_{z \in \{1,i,j,k\}} E^{(z)^2}_{2\alpha}.\end{equation}

This differential operator $\Omega$ coincides with the infinitesimal action of the Casimir element of $\mathfrak{g}_0$ on the space of right $K$-invariant smooth functions of $G/Z^+$. In what follows, we shall refer to it as the Casimir operator.

\section{Automorphic forms}
For $\lambda \in \C$ and a discrete subgroup $\Gamma \in \SL_2(\R)$, we denote by $S(\Gamma,\lambda)$ the space of Maass cusp forms of weight $0$ on the complex upper half plane $\mathfrak{h}$ whose eigenvalue with respect to the hyperbolic Laplacian is $-\lambda$.

For a discrete subgroup $\Gamma \subset \GL_2(\mathbb{H})$ and $r \in \C$, let $\mathcal{M}(\Gamma,r)$ denote the space of smooth functions $F$ on $\GL_2(\mathbb{H})$ which satisfy the following conditions : 
\begin{enumerate}
	\item $\Omega\cdot F = -\frac{1}{2}\Big(\frac{r^2}{4}+ 1\Big)F$, where $\Omega$ is the Casimir operator defined in \eqref{Casimir},
	
	\item for any $(z,\gamma, g, k) \in Z^+ \times \Gamma \times G \times K$, we have $F(z\gamma g k) = F(g)$,
	
	\item F is of moderate growth.
	
\end{enumerate}

For automorphic forms of $\SL_2(\R)$ we will concern ourselves only with the congruence subgroup $\Gamma_0(2) \in \SL_2(\R)$ of level 2. For the choice of a discrete subgroup of $\GL_2(\HH)$, note that the definite quaternion algebra $B$ has a unique maximal order $\mathcal{O}$ given by: 
$$ \mathcal{O} = \Z + \Z i + \Z j + \Z \frac{1 + i+ j+ij}{2},$$ 
called the Hurwitz order. The discrete subgroup we shall consider in this case will be $\GL_2(\mathcal{O})$.

We denote by
\begin{equation} \label{Sdef}
\mathcal{S} \coloneqq \Z(1 -ij) + \Z  (-i- ij) + \Z  (-j - ij) + \Z (2ij)
\end{equation}
 the dual lattice of $\mathcal{O}$ with respect to the bilinear form on $\HH \times \HH$ defined by Re $= \frac{1}{2} tr$. We denote by $\varpi_2 \coloneqq (1+i)$ which is the uniformizer of $B \otimes_\Q \Q_2$.
 
\begin{lem}
We have $\mathcal{S} = \varpi_2 \mathcal{O}$
\end{lem}

In terms of $\mathcal{S}$, any $F \in \MG$ has a Fourier expansion of the form 

\begin{equation}
F(n(x)a_y) = u(y) + \sum_{\beta\in S\setminus \{0\}}A(\beta)y^2K_{\sqrt{-1}r} (2\pi|\beta|y)e^{2\pi\sqrt{-1}\text{Re}(\beta x)}.
\end{equation}
Here $K_\alpha$, with $\alpha \in \C$, is the modified Bessel function, which satisfies the differential equation 
$$y^2\frac{d^2K_\alpha}{dy^2} + y\frac{dK_\alpha}{dy} -(y^2+\alpha^2)K_\alpha = 0.$$
We call $F$ cuspidal if $u(y)=0$.
\chapter{Lifting to GL(2) over a division quaternion algebra by Muto, Narita and Pitale}

\section{Construction of lift}
We first define the set of primitive elements of $\mathcal{S}$, denoted $\mathcal{S}^{prim}$, by 
\begin{equation}
\mathcal{S}^{prim} \coloneqq \{\beta \in \mathcal{S} \setminus \{0\} \mid \varpi_2 | \beta, \varpi_2 ^2 \nmid \beta, n \nmid \beta \text{ for all odd integers n} \}. \label{3.1}
\end{equation}
Any $\beta \in \mathcal{S}\setminus \{0\}$ can then be uniquely written as 
\begin{equation}
\beta = \varpi_2^u \cdot n \cdot \beta_0, \label{3.2}
\end{equation}
where $u$ is a non-negative integer, $n$ is an odd positive integer and $\beta_0 \in \mathcal{S}^{prim}$.

Let $c(N)$ be the  Fourier coefficients of $f \in \SG$.  Assuming it is an Atkin-Lehner eigenfunction with eigenvalue $\epsilon \in \{\pm 1\}$, set 
\begin{equation}
A(\beta) \coloneqq |\beta|\sum_{t=0}^{u} \sum_{d \mid n}(-\epsilon)^t c\Big(-\frac{|\beta|^2}{2^{t+1}d^2}\Big). \label{3.3}
\end{equation}

With $A(\beta)$ as above, Muto, Narita and Pitale prove the following theorem.
\begin{thm}[Theorem 4.4 in \cite{MNP}] \label{Pitale}
	Let $f \in \SG$ with Fourier coefficients $c(N)$ and eigenvalue $\epsilon$ of the Atkin-Lehner involution. Define
	\begin{equation}
	\F(n(x)a_y) \coloneqq \sum_{\beta \in \Ss \setminus \{0\}} A(\beta)y^2K_{\sqrt{-1}r}(2\pi|\beta|y)e^{2\pi
		\sqrt{-1}Re(\beta x)}
	\end{equation}
	with $\{A(\beta)\}_{\beta \in \Ss \setminus \{0\}}$ as in (\refeq{3.3}). 
	Then we have $\F \in \M (\GL_2(\mathcal{O}), r)$ and $\F$ is a cusp form. 
	Furthermore, $\F \not \equiv 0$ if $f \not \equiv 0.$ 
\end{thm}

The fundamental tool used in the proof is the following converse theorem of Maass \cite{M2}.

\begin{thm}[Maass]
Let $\{A(\beta)\}_{\beta \in S\setminus \{0\}}$ be a sequence of complex numbers such that 
$$A(\beta) = O(|\beta|^\kappa) \hspace{1 in} (\exists \kappa > 0)$$
and put 
$$F(n(x)a_y) \coloneqq \sum_{\beta \in \Ss \setminus \{0\}} A(\beta)y^2K_{\sqrt{-1}r}(2\pi|\beta|y)e^{2\pi
	\sqrt{-1}Re(\beta x)}.$$
For a Harmonic polynomial $P$ on $\HH$ of degree $l$ we introduce
$$\xi(s,P) \coloneqq \pi^{-2s}\Gamma\left(s+\frac{\sqrt{-1}r}{2}\right)\Gamma\left(s-\frac{\sqrt{-1}r}{2}\right)\sum_{\beta \in S \setminus\{0\}}A(\beta)\frac{P(\beta)}{|\beta|^{2s}},$$
which converges for $Re(s) > \frac{l+4+\kappa}{2}$. Let $\{P_{l,\nu}\}_\nu$ be a basis of Harmonic polynomials on $\HH$ of degree $l$. 

Then $F\in \mathcal{M}(\Gamma_T,r)$ is equivalent to the condition that, for any $l,\nu$, the $\xi(s,P_{l,\nu})$ satisfies the following three conditions.
\begin{enumerate}
\item It has analytic continuation to the whole complex plane.

\item It is bounded on any vertical strip of the complex plane.

\item The functional equation 
$$\xi(2+l-s,P_{l,\nu})=(-1)^l\xi(s,\hat{P}_{l,\nu})$$
holds, where $\hat{P}(x) \coloneqq P(\bar{x})$ for $x \in \HH$.
\end{enumerate}
\end{thm}

Here, $\Gamma_T$ is the subgroup of $\GL_2(\mathcal{O})$ generated by 
$$ \begin{bmatrix}
0&-1\\
1&0 \\
\end{bmatrix},
\begin{bmatrix}
1&\beta\\
0&1\\
\end{bmatrix} \quad
(\beta\in \mathcal{O}).$$

\section{Action of the Hecke operators}
Let $\mathcal{G}(\A) = \GL_2(B_\A)$, where $B_\A$ denotes the adelization of $B$, and let U be the compact subgroup of $\mathcal{G}(\A)$ given by $\Pi_{p < \infty} \GL_2(\mathcal{O}_p)$, where $\mathcal{O}_p$ denotes the p-adic completion of $\mathcal{O}$ at a finite prime $p$. For a complex number $r \in \C$, the space of automorphic forms for $\mathcal{G}$, denoted $M(\mathcal{G}(\A),r)$, is defined as follows.

\begin{definition}\label{Def3.1}
Let $M(\mathcal{G}(\A),r)$ be the space of smooth functions $\Phi$ on $\mathcal{G}(A)$ satisfying the following conditions:
\begin{enumerate}
\item $\Phi(z \gamma g u_f u_\infty) = \Phi(g)$ for any $(z, \gamma, g, u_f, u_\infty) \in Z_\A \times \mathcal{G}(\Q) \times \mathcal{G}(\A) \times U \times K$, where $Z_\A$ denotes the center of $\mathcal{G}(\A)$,

\item $\Omega \cdot \Phi (g_\infty) = -\frac{1}{2}\left( \frac{r^2}{4}+1\right)\Phi(g_\infty)$ for any $g_\infty \in \mathcal{G}(\R) = \GL_2(\HH)$,

\item $\Phi$ is of moderate growth.
\end{enumerate}
\end{definition}
The class number of $\mathcal{G}$ with respect to $U$ is one, which means that $\mathcal{G}(\A) = \mathcal{G}(\Q)\mathcal{G}(\R)U$. We can thus view $F\in \M(\GL_2(\mathcal{O}),r)$ as a smooth function $\Phi_F$ on $\mathcal{G}(\A)$ given by 
$$\Phi_F(\gamma g_\infty u_f) = F(g_\infty) \quad \forall(\gamma, g_\infty, u_f) \in \mathcal{G}(\Q) \times \mathcal{G}(\R) \times U.$$
Hence, $M(\mathcal{G}(\A),r) \simeq \mathcal{M}(\GL_2(\mathcal{O}),r)$.

Section 5 of \cite{MNP} proves that if $f$ is a Hecke eigenform, then $F_f$ is also a Hecke eigenform. For each place $p \leq \infty$ let $\mathcal{G}_p \coloneqq \GL_2(B_p)$ for $B_p = B \otimes_\Q \Q_p$. For any finite prime $p \neq 2$, we have $\GL_2(B_p) \simeq \GL_4(\Q_p)$. Let $\mathcal{O}_p$ be the p-adic completion of $\mathcal{O}$ as above. Then, for $p \neq 2$, $\mathcal{O}_p \simeq M_2(\Z_p)$ and $\GL_2(\mathcal{O}_p) \simeq \GL_4(\Z_p)$. Set $K_p = \GL_2(\mathcal{O}_p)$ for all $p < \infty$.

According to \cite{Sat}, the Hecke algebra of $\GL_2(B_p)$ with respect to $\GL_2(\mathcal{O}_p) $ is generated by: 
\[
\begin{cases}
\{\varphi_1^{\pm 1}, \varphi_2 \} & \text{ if } p=2\\
\{\phi_1^{\pm 1}, \phi_2,\phi_3, \phi_4\} & \text{ if } p\neq 2.
\end{cases}
\]
Here $\varphi_1,\varphi_2$ denote the characteristic functions of 
$$ K_2 \begin{bmatrix}
\varpi_2 & 0 \\
0 & \varpi_2 \\
\end{bmatrix}K_2,
K_2 \begin{bmatrix}
\varpi_2 & 0 \\
0 & 1 \\
\end{bmatrix}K_2 $$ 
respectively, whereas $\phi_1,\phi_2,\phi_3,\phi_4$ denote the characteristic functions of $K_p h_i K_p$ where $h_i, 1 \leq i\leq 4$ are
$$\begin{bmatrix}
p & & &\\
& p & &\\
& & p &\\
& & & p
\end{bmatrix},
\begin{bmatrix}
p & & &\\
& p & &\\
& & p &\\
& & & 1
\end{bmatrix},
\begin{bmatrix}
p & & &\\
& p & &\\
& & 1 &\\
& & & 1
\end{bmatrix},
\begin{bmatrix}
p & & &\\
& 1 & &\\
& & 1 &\\
& & & 1
\end{bmatrix}$$
respectively for $p \neq 2$.
We will define the set $ C_p \coloneqq \{\alpha \in \mathcal{O} \mid \nu(\alpha) = p \}/ \mathcal{O}^\times$. The following Proposition 5.8 from \cite{MNP} allows us to explicitly compute the action of the Hecke operators on the Fourier coefficients $A(\beta)$ of $F \in \MG$. 

\begin{prop}[ Proposition 5.8 from \cite{MNP}] \label{prop3.1}
	1. Let $p = 2$ and $h =
	\begin{bmatrix}
	\varpi_2 & 0\\
	0 & 1
	\end{bmatrix}$. We obtain
	$$(K_2hK_2 \cdot F)_\beta = 2(A(\beta \varpi_2^{-1}) + A(\beta \varpi_2)).$$
	2. Let $p$ be an odd prime and $\beta \in \mathcal{S}\setminus \{0\}$.\\
	(a) When h =
	$\begin{bmatrix}
	p & & &\\
	& p & &\\
	& & p &\\
	& & & 1
	\end{bmatrix}$,
	$$(K_phK_p \cdot F)_\beta = p( \sum_{\alpha \in C_p} A(\beta \bar{\alpha}^{-1}) + \sum_{\alpha \in C_p} A(\bar{\alpha}\beta)).$$\\
	(b) When h = $\begin{bmatrix}
	p & & &\\
	& 1 & &\\
	& & 1 &\\
	& & & 1
	\end{bmatrix}$, 
	$$(K_phK_p \cdot F)_\beta = p( \sum_{\alpha \in C_p}A(\alpha^{-1} \beta) + \sum_{\alpha \in C_p}A(\beta \alpha)).$$\\
	(c) When h =
	$\begin{bmatrix}
	p & & &\\
	& p & &\\
	& & 1 &\\
	& & & 1
	\end{bmatrix}$, 
	$$(K_phK_p \cdot F)_\beta = (p^2 A(p^{-1}\beta ) + p^2A(p\beta) + p \sum_{(\alpha_1,\alpha_2) \in C_p \times C_p} A(\alpha_1^{-1} \beta \alpha_2)).$$
\end{prop}


\section{Automorphic representation corres-\\ponding to the lifting}
The above action of the Hecke algebra allows us to find the Hecke eigenvalues for $\F$ in terms of the Hecke eigenvalues of $f$. 

\begin{prop} 
	\label{Prop3.2}
	Let $f \in \SG$ be a Hecke eigenform with eigenvalue $\lambda_p$ for every odd prime $p$ and the Atkin-Lehner eigenvalue $\epsilon$. Then $F = \F$ as defined in Theorem \ref{Pitale} is a Hecke eigenform with eigenvalues $\tensor*[_p]{\mu}{_1}, \tensor*[_p]{\mu}{_2}, \tensor*[_p]{\mu}{_3}, \tensor*[_p]{\mu}{_4}$ for $\phi_1, \phi_2,\phi_3, \phi_4$ respectively at every odd prime $p$ and $_2\mu_1$, $\tensor*[_2]{\mu}{_2}$ for $\varphi_1, \varphi_2$ at $p=2$. They are related as
	\begin{equation} \tensor*[_p]{\mu}{_1} = 1 , \tensor*[_p]{\mu}{_2} = \tensor*[_p]{\mu}{_4} = p(p+1)\lambda_p, \tensor*[_p]{\mu}{_3} = p^2\lambda_{p}^{2} + p^3 + p \label{3.5}
	\end{equation}
	and 
	\begin{equation}
	\tensor*[_2]{\mu}{_1} = 1, \tensor*[_2]{\mu}{_2} = -3\sqrt{2}\epsilon. \label{3.6}
	\end{equation}
\end{prop}

This is proved in Proposition 5.9, 5.11 and 5.12 of \cite{MNP}. Let the representation $\pi_F$ denote the irreducible cuspidal automorphic representation of $\GL_2(B_\A)$ corresponding to $\F$ with $B_\A \coloneqq \otimes_{p \leq \infty}'B_p$ generated by right translates of $\Phi_F$ (as in Definition \ref{Def3.1}). $\pi_F$ is cuspidal as $\F$ is a cusp form and the irreducibility follows from the strong multiplicity-one result for $\GL_2(B_\A)$ (c.f. \cite{5},\cite{4}). Let $\pi_F = \otimes'_p \pi_p$, where $\pi_p$ an irreducible admissible representation of $\GL_2(B_p)$ for $p<\infty$ and $\pi_\infty$ is an irreducible admissible representation of $\GL_2(\HH)$. 

Let $\mathcal{B}_2$ and $\mathcal{B}_4$ denote the group of upper triangular matrices in $\GL_2$ and $\GL_4$ respectively. Then, for $p < \infty$ and odd, 
$\pi_p$ is the unique spherical constituent of the unramified principal series representation $\text{Ind}_{\mathcal{B}_4(\Q_p)}^{\GL_4(\Q_p)}(\chi_1 \times \chi_2 \times \chi_3 \times \chi_4)$
where $\chi_i$ are unramified character of $\Q_p^\times$ given by
\begin{align*}
\chi_1(p) = p^{1/2}\frac{\lambda_p + \sqrt{\lambda_p^2 -4}}{2}, \hspace{0.1 in} & \chi_2(p) = p^{1/2}\frac{\lambda_p - \sqrt{\lambda_p^2 -4}}{2},\\ 
\chi_3(p) = p^{-1/2}\frac{\lambda_p + \sqrt{\lambda_p^2 -4}}{2}, \hspace{0.1 in} & \chi_4(p) = p^{-1/2}\frac{\lambda_p - \sqrt{\lambda_p^2 -4}}{2}. \numberthis
\end{align*}

For $p =2$, $\pi_2$ is the unique spherical constituent of the unramified
principal series representation $\text{Ind}_{\mathcal{B}_2(\Q_2)}^{\GL_2(B_2)}(\chi_1 \times \chi_2)$ with

\begin{equation}
\chi_1(\varpi_2) = -\sqrt{2}\epsilon, \chi_2(\varpi_2) = -1/\sqrt{2}\epsilon.
\end{equation}

At the prime $p =\infty$, the archimedian component $\pi_\infty$ is isomorphic to the principal series $\text{Ind}^{\GL_2(\mathbb{H})}_{\mathcal{B}_2(\HH)}(\chi_{\pm \sqrt{-1}r/2})$ with 

\begin{equation}
\chi_s\left(\begin{bmatrix}
a & *\\
0 & d
\end{bmatrix} \right) = \nu(ad^{-1})^s.
\end{equation}

These local representations are explicitly constructed in Section 6 of \cite{MNP}.

\chapter{Maass space in $\MG$}
\section{Definition of the Maass space}
We will call the image of the lift constructed in Theorem \ref{Pitale} to $\M(\GL_2(\mathcal{O}),r)$ as the Maass space. To characterize the functions in the Maass space, we first define the following subspace.

\begin{dfn} \label{rec}
	Let $\Ms$ denote the subspace of cusp forms $F$ in $\MG$ Fourier coefficients $A(\beta)$ satisfy:
	\begin{enumerate}
		\item If $\beta = \varpi_2^u n\beta_0 $ as in (\ref{3.2}), then $A(\beta)$ depend only on $K = |\beta|^2$, $u$  and $n$. We shall then write $A(\beta)$ as $A(K,u,n)$.
		
		\item $A(K,u,n)$ satisfy the recurrence relations : 
		\begin{enumerate}
			
			\item $A(K,u,n) = \frac{-3\epsilon}{\sqrt{2}}A(\frac{K}{2},u-1,n) - A(\frac{K}{4},u-2,n)$ for some $\epsilon \in \{\pm 1\}$ \label{2a},
			
			\item  $A(K,u,n)= \sum_{d \mid n} d \cdot A\left(\frac{K}{d^2},u,1\right)$. \label{2b}
		\end{enumerate}	
	\end{enumerate}
\end{dfn}

We will define $A(K,u,n) = 0$ if $u$ is negative. These recurrence relations are similar to those of Maass in the case of Saito-Kurokawa lifts in \cite{Maass}. 

\section{Isolating c(-N)}
All our information about the Fourier coefficients $\{A(\beta)\}_{\beta \in S \setminus \{0\}}$ of $\F$ is obtained from the Fourier coefficients $c(-N)$ of $f$ from equation \eqref{3.3}. To do any successful manipulation of $A(\beta)$, we would ideally like to have a formula for $c(-N)$ in terms of the Fourier coefficients $A(\beta)$. 

\begin{prop} \label{prop1}
	Let $f \in \SG$ be an Atkin-Lehner eigenform with eigenvalue $\epsilon \in \{\pm 1 \}$ and which is a Hecke eigenform at $p=2$. We will denote by the Fourier coefficients of $f$ by $\{c(N)\}$. Let $\F$ be as in Theorem \ref{Pitale}. For $N = 4^a b$ with $a,b$ non-negative integers and $4 \nmid b $, we get
	\begin{equation}
	c(-N) = \frac{A(2N,u,1)}{\sqrt{2N}} + \epsilon \frac{A(N,u-1,1)}{\sqrt{N}} \label{4.1}
	\end{equation}
	where \[ 
	u = 
	\begin{dcases}
	2a & \text{if }b \equiv 1,3 \mod (4) \\
	2a+1 & \text{if }b \equiv 2 \mod (4) \\ 
	\end{dcases}
	\]
\end{prop}

Note that there is more than one $\beta$ with the same $K,u$ and $n$. However, by construction in equation (\ref{3.3}) all such $\beta$ give the same $A(\beta)$. As such, $c(-N)$ is well defined in terms of $\beta$ representatives of $A(K,u,n)$. We will need the following lemmas for the proof of Proposition \ref{prop1}.

\begin{lem} \label{lem1}
	$\beta = (x +yi+ zj +w ij) \in \mathcal{S}^{prim}$ iff $|\beta|^2 \equiv 2 \mod 4$ and $\gcd(\beta) \coloneqq \gcd(x,y,z,w)=1$.		
\end{lem}

\begin{proof}
	First, we prove the following claim.
	
\begin{claim}
	Let $\beta \in \mathcal{O}$. Then $\beta \in \Ss$ iff $|\beta|^2 \equiv 0, 2\mod 4$.
\end{claim}
\begin{proof}[Proof of claim]
Simplifying the condition from (\ref{Sdef}), we see that $x+y+z+w \equiv 0 \mod 2$ and therefore $x^2 + y^2 +z^2 +w^2 \equiv 0 \mod 2$ or equivalently $x^2 + y^2 +z^2 +w^2\equiv 0 ,2 \mod 4$. If $\beta \in \mathcal{O}$ such that $|\beta|^2 \equiv 0 \mod 2$ then by parity conditions $x+y+z+w \equiv 0 \mod 2$ implying $\beta \in \mathcal{S}$. Hence, $\beta \in S$ iff $|\beta|^2 \equiv 0, 2\mod 4$.
\end{proof}			
	Now, consider an element $\beta_1 \in \mathcal{S}$ with $\gcd(\beta_1)=1$ such that $\beta_1 \notin \mathcal{S}^{prim}$. Then by definition of $\mathcal{S}^{prim}$ in (\ref{3.1}) this means $\beta_1 = \varpi_2^2\beta$ for some $\beta \in \mathcal{O}$. Now, $|\varpi_2^2|^2 = 4$ hence, $4| |\beta_1|^2$ implying $|\beta_1|^2 \equiv 0 \mod 4$. 
	
	Conversely, for $\beta \in \Ss$ if $|\beta|^2 \equiv 0 \mod 4$ then $|\varpi_2^{-1}\beta|^2 \equiv 0,2 \mod 4$. Since $\varpi_2^{-1} = \frac{1-i}{2}$ it is an easy verification that $\varpi_2^{-1}\beta \in \mathcal{O}$. Then by claim, $\varpi_2^{-1}\beta \in \mathcal{S}$. Therefore, $\beta \in \varpi_2\mathcal{S}$ which is to say $\beta \notin \Sp$.
	
	Therefore, $\beta \in \mathcal{S}$ satisfies $|\beta|^2 \equiv 2 \mod 4$ with $\gcd(\beta) =1$ iff $\beta \notin \varpi_2 S$ and equivalently $\beta \in \mathcal{S}^{prim}$ as required. \end{proof}

For any $N$, the easiest way for there to exist a $\beta$ with $\gcd(\beta)=1$ and $|\beta|^2 = 2N$ is if $w = 1$ with $x^2 + y^2 +z^2 = 2N -1$. By Legendre's three square Theorem, an odd number $2N-1$ cannot be written as a sum of three squares iff $2N-1 \equiv 7 \mod 8 \Leftrightarrow 2N \equiv 0 \mod 8 \Leftrightarrow N \equiv 0 \mod 4$. However, if $x^2 +y^2+z^2 = 2N-1$ and $x,y,z$ are all odd then $4|(x^2+y^2+z^2+1)$ and hence $\beta \notin \Ss^{prim}$.

\begin{lem} \label{lem2}
	If $N \equiv 1,3 \mod 4$, then there is a $\beta \in \Ss^{prim}$ such that
	\begin{equation}
	c(-N) = \frac{A(\beta)}{\sqrt{2N}}.
	\end{equation}
\end{lem}

\begin{proof}[Proof of Lemma \ref{lem2}]
	If $N \equiv 1,3 \mod 4$ then $2N -1 \equiv 1,5 \mod 8$ respectively. Therefore, by Legendre's theorem, there exist $ x,y,z \in \Z$, not all odd, such that $2N-1 = x^2+y^2+z^2$. Hence, by Lemma \ref{lem1}, $\beta = (x + yi + zj + ij)\in \Ss^{prim}$ and $\beta = \varpi_2^0 \cdot 1 \cdot \beta$. Then (\ref{3.3}) becomes
	$$A(\beta) = |\beta| c\Big(\frac{-|\beta|^2}{2}\Big) = \sqrt{2N}c(-N).$$
	Solving for $c(-N)$ gives the required result.
\end{proof}

If $N \equiv 2 \mod 4$, we can still find $x,y,z \in \Z$ such that $x^2 +y^2+z^2 +1^2= 2N$. As $|\beta|^2 = 2N \equiv 0 \mod 4$, $\beta \in \Ss$ but $\beta \notin \Sp$ by Lemma \ref{lem1}. However, $\varpi_2^{-2} \beta \notin \Ss$ as $|\varpi_2^{-2} \beta|^2 \equiv 1 \mod 4$. Therefore $\beta = \varpi_2^1 \cdot 1 \cdot \beta_0$ for some $\beta_0 \in \Ss^{prim}$.

If $N \equiv 0 \mod 4$ then one cannot find $x,y,z \in \Z$ such that $x^2+y^2+z^2 =2N$. In that case, write $N = 4^a b$ with $4 \nmid b$. Then, we can find $x,y$ and $z$ such that $x^2 +y^2 +z^2 + 1 = 2b$. 
This allows us to find $\beta\in \Ss$ such that $|\beta|^2 = 2N$ and $\beta = \varpi_2^{2a} \cdot 1 \cdot \beta_0$ if $b \equiv 1,3 \mod 4$ and $\beta = \varpi_2^{2a+1} \cdot 1 \cdot \beta_0$ if $b \equiv 2 \mod 4$ with $\beta_0 \in \Sp$.

\begin{lem}\label{lem3}
	If $N \equiv 0,2  \mod 4$, then $\exists \beta \in \Ss$ such that
	\begin{equation}
	c(-N) = \frac{A(\beta)}{\sqrt{2N}} + \epsilon \frac{A( \varpi_2^{-1}\beta)}{\sqrt{N}}.
	\end{equation}
\end{lem}
\begin{proof}[Proof of Lemma \ref{lem3}]
	Let $\beta = \varpi_2^u \cdot 1 \cdot \beta_0 \in \Ss$ with $|\beta|^2 = 2N$ as in the remark before the lemma. 
	Unlike Lemma \ref{lem2}, in this case $u \neq 0$. Following (\ref{3.3}), we get
	\begin{align*}
A(\beta) = & |\beta|\sum_{t=0}^u (-\epsilon)^tc\left(-\frac{|\beta|^2}{2^{t+1}}\right)\\
 = & \sqrt{2N}c(-N) +\sqrt{2N}\sum_{t=1}^u (-\epsilon)^tc\left(-\frac{|\beta|^2}{2^{t+1}}\right)
	\end{align*} 
	But \begin{align*}
	\sum_{t=1}^u (-\epsilon)^t c\left(-\frac{|\beta|^2}{2^{t+1}}\right) 
	&= \frac{-\epsilon}{\sqrt{N}} \sqrt{N}\sum_{t=0}^{u-1} (-\epsilon)^t c\left(-\frac{|\beta|^2 / 2}{2^{t+1}}\right) 
	\\&= \frac{-\epsilon}{\sqrt{N}} \sqrt{N}\sum_{t=0}^{u-1} (-\epsilon)^t c\left(-\frac{|\varpi_2^{-1}\beta|^2 }{2^{t+1}}\right) 
	\\&= \frac{-\epsilon}{\sqrt{N}} A(\varpi_2^{-1}\beta).
	\end{align*}
	
	Rearranging the terms gives us the required result.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop1}]
	Let $N = 4^a b$ with $a$ and $b$ as in the statement of the proposition. 
	If $b \equiv 1,3 \mod 4$ then by Lemma \ref{lem2}, we can find $\beta' \in \Sp$ such that $|\beta'|^2 = 2b$. Then $\beta = \varpi_2^{2a} \beta'$ has $|\beta|^2 = 2N$, $u=2a$ and $n=1$. Hence, in this case $A(\beta) = A(2N, 2a, 1)$. Then $A(\varpi_2^{-1}\beta) = A(N,2a-1,1)$ and the first condition of the proposition follows from Lemma \ref{lem3}. If $a =0$ then we are in case of Lemma \ref{lem2} with $A(N,-1,1) =0$.
	
	If $b \equiv 2 \mod 4$ then we can find $\beta' = \varpi_2 \cdot 1 \cdot \beta_0 \in \Ss$ such that $\beta = \varpi_2^{2a} \beta'$ satisfies $A(\beta) = A(2N, 2a+1,1)$ and $A(\varpi_2^{-1}\beta) = A(N,2a,1)$. The second condition of the proposition now follows from Lemma \ref{lem3}.
\end{proof}


\section{First result}

Now that we have an explicit formula for the Fourier coefficients $c(-N)$ in terms of the Fourier coefficients $A(K,u,n)$ of $F_f$, we can prove our first result towards identifying the image of the lift.

\begin{thm} \label{Th1}
	Let $f \in \SG$ be an Atkin-Lehner eigenform with eigenvalue $\epsilon \in \{\pm 1 \}$ and which is a Hecke eigenform at $p=2$. Then $\F$ obtained in Theorem \ref{Pitale} belongs to the subspace $\M^*(\GL_2(\mathcal{O}),r)$.
\end{thm}

\begin{proof}
	
	The Fourier coefficients of $F_f$ are given in terms of the Fourier coefficients of $f$ as in equation (\ref{3.3}) by: 
	$$A(\beta) = |\beta|\sum_{t=0}^{u} \sum_{d \mid n}(-\epsilon)^t c\Big(-\frac{|\beta|^2}{2^{t+1}d^2}\Big).$$
	The only properties of $\beta$ used in here are $|\beta|, u$ and $n$. Replacing $|\beta|$ with $|\beta|^2$ we can say that the Fourier coefficients $A(\beta)$ of $F_f$ are depend only on $|\beta|^2, u$ and $n$, satisfying the first condition of Definition \ref{rec}. 
	
	To prove equation \eqref{2b}, we use the value of $c(-N)$ from \eqref{4.1} and substitute $A(K,u,n)$ for $A(\beta)$. Doing so, we get:
	\begin{align*}
	A(K,u,&n)= 
	\sqrt{K} \sum_{t=0}^u \sum_{d \mid n} (-\epsilon)^t c\left(-\frac{K}{2^{t+1}d^2}\right) 
	\\=& \sqrt{K} \sum_{t=0}^u \sum_{d \mid n} (-\epsilon)^t \Bigg(\frac{A(K/(2^{t}d^2),u-t,1)}{\sqrt{K/(2^{t}d^2)}}\\ 
	&+ \epsilon \frac{A(K/(2^{t+1}d^2),u-t-1,1)}{\sqrt{K/(2^{t+1}d^2)}}\Bigg)
	\\=&\sqrt{K} \sum_{d \mid n} \Bigg(\sum_{t=0}^u \Bigg( (-\epsilon)^t	 \frac{A((K/d^2)/2^{t},u-t,1)}{\sqrt{(K/d^2)/2^{t}}} \\
	&- (-\epsilon)^{t+1} \frac{A((K/d^2)/2^{t+1},u-(t+1),1)}{\sqrt{(K/d^2)/2^{t+1}}}\Bigg)\Bigg)
	\\=& \sqrt{K} \sum_{d \mid n} \frac{A(K/d^2,u,1)}{\sqrt{K/d^2}}
	\\=& \sum_{d \mid n} d A(K/d^2,u,1)
	\end{align*}
	
	For equation \eqref{2a}, note that $F_f$ is Hecke eigenform by Proposition 5.9 of \cite{MNP} for $p=2$ since we have assumed the same for $f$. Then, by Proposition 5.10 of \cite{MNP}, the Fourier coefficients of the lift $F_f$ satisfy
	$$2(A(\beta \varpi_2) + A(\beta \varpi_2^{-1})) = -3\sqrt{2}\epsilon A(\beta)$$ with $A(\beta \varpi_2^{-1}) = 0$ if $u=0$.
	Writing it in terms of $K,u$ and $n$, we get 
	$$2(A(2K,u+1,n) + A(K/2,u-1,n)) = -3\sqrt{2}\epsilon A(K,u,n)$$ or equivalently
	
	$$A(K,u,n) = \frac{-3\epsilon}{\sqrt{2}}A(\frac{K}{2},u-1,n) - A(\frac{K}{4},u-2,n)$$
	for $u \geq 1$ with $A(\frac{K}{4},u-2,n)= 0$ for $u=1$.
\end{proof}

We now wish to prove the converse of Theorem \ref{Th1} which is to show that our `necessary' condition is also `sufficient'. We do so first in Theorem \ref{Thm5.1} for the case of Hecke eigenforms and then in Theorem \ref{MainTh} for the general case.

\chapter{The Jacquet-Langlands correspondence for $\GL(2,B) \leftrightarrow \GL(4)$}

\section{Description of the automorphic representation }
If $F$ is a cuspidal Hecke eigenform, let the automorphic representation associated with it be denoted by $\Pi_F \simeq \otimes'_{p \leq \infty}\Pi_{F,p}$. At every prime $p$, the local component $\Pi_{F,p}$ is a spherical representation of $\GL_2(B_p)$ with $B_p = B \otimes \Q_p$. The representation is cuspidal since the Hecke eigenform $F$ is cuspidal. 

For every odd $p < \infty$ we have $\GL_2(B_p) \cong \GL_4(\Q_p)$. From Section 5.2 and 6.1 of \cite{MNP}, we have $\Pi_{F,p}$ is the unique irreducible constituent of some unramified principal series representation $\ind_{\mathcal{B}_4(\Q_p)}^{\GL_4(\Q_p)}(\chi_1 \times \chi_2 \times \chi_3 \times \chi_4)$ where each $\chi_i$ is an unramified character of $\Q_p$. 

For $F \in \Ms$, the characters $\chi_1,\chi_2,\chi_3,\chi_4$ have a special form. This is proved in the next proposition.

\begin{prop}\label{prop5.1}
	For every odd prime $p$, there is a $\lambda_{p} \in \C$ such that, up to the action of the Weyl group, $\chi_i$ are given by the formula
	\begin{align*} 
	\chi_1(p) = p^{1/2}\frac{\lambda_p + \sqrt{\lambda_p ^2 - 4}}{2} &; \quad \chi_2(p) = p^{1/2}\frac{\lambda_p - \sqrt{\lambda_p ^2 - 4}}{2};\\
	\chi_3(p) = p^{-1/2}\frac{\lambda_p + \sqrt{\lambda_p ^2 - 4}}{2} &; \quad \chi_4(p) = p^{-1/2}\frac{\lambda_p - \sqrt{\lambda_p ^2 - 4}}{2} \numberthis \label{5.1}
	\end{align*}
\end{prop}

The proof of the proposition will use \cite[Lemma~5.10]{MNP}. 
\begin{lem}[{\cite[Lemma~5.10]{MNP}}] \label{l51}
	Let $\beta \in \Sp$. Then 
	
	$$\#\{\alpha \in C_p : p| \beta \alpha \} = \#\{ \alpha \in C_p : p|\alpha\beta\}
	= \begin{cases} 
	1 \quad & \text{if } p\mid |\beta|^2, \\
	0 \quad & \text{if } p\nmid |\beta|^2\\
	\end{cases}$$
	In addition, $p^2$ does not divide $\alpha \beta$ or $\beta \alpha$ for any $\alpha \in C_p$
\end{lem} 
Here $ C_p \coloneqq \{\alpha \in \mathcal{O} | \nu(\alpha) = p \}/ \mathcal{O}^\times$ with $\#(C_p) = (p+1)$.
In terms of $A(K,u,n)$, if $A(\beta) = A(K,0,1)$ with $p\nmid K$ then $A(\alpha \beta) = A(\beta \alpha) = A(pK,0,1)$ for every $\alpha \in C_p$. If $p\mid K$ then there are unique $\alpha_1,\alpha_2 \in C_p$ (not necessarily different) such that $A(\alpha_1 \beta) = A(\beta \alpha_2) = A(pK,0,p)$ and $A(\alpha \beta) = A(\beta \alpha) = A(pK,0,1)$ in every other case.

\begin{proof}[Proof of Proposition \ref{prop5.1}]
	It is enough to show that the Hecke eigenvalues $\tensor*[_p]{\mu}{_1}, \tensor*[_p]{\mu}{_2}$, $\tensor*[_p]{\mu}{_3}, \tensor*[_p]{\mu}{_4}$ for $F$ satisfy the equation (\ref{3.5}). The fact that this is enough follows from the proof of Proposition 6.2 from \cite{MNP}.
	We will follow notation of Proposition \ref{prop3.1} for the Hecke algebra and refer to diagonal matrices given before it by $h_2, h_3$ and $h_4$ respectively. 
	
	Since the Maass form $F$ is non-zero, at least one of the Fourier coefficients $A(K,u,n)$ is non-zero. This implies, from the recurrence conditions of Definition \ref{rec}, that there exists at least one $K$ such that $A(K,0,1) \neq 0$. Let $K = p^nK_0$ where $K_0$ is co-prime to $p$. Then we claim that
	\begin{equation} \label{lambda p}
	\lambda_p = \frac{A(p^{n+1}K_0,0,1) + A(p^{n-1}K_0,0,1)}{ A(p^nK_0,0,1)}
	\end{equation} with $A(p^{n-1}K_0,0,1) = 0$ if $n=0$.
	
	\textbf{Case 1: }$K=p^0K_0$ i.e. to say $p \nmid K$. Let $\beta$ such that $A(\beta) = A(K,0,1)$. By Lemma \ref{l51}, for every $\alpha \in C_p$, we have $\beta \alpha^{-1} \notin \mathcal{O}$ and $p\nmid \overline{\alpha}\beta$. Therefore, $A(\beta \bar{\alpha}^{-1}) = 0$ and $A(\bar{\alpha}\beta) = A(pK,0,1)$ for every $\alpha \in C_p$.
	Then, condition 2(a) of Proposition \ref{prop3.1} implies
	\begin{align*}
	(K_p h_2 K_p \cdot F)_{\beta} = & p\big( \sum_{\alpha \in C_p} A(\beta \bar{\alpha}^{-1}) + \sum_{\alpha \in C_p} A(\bar{\alpha}\beta)\big)\\
	= & p\big(\sum_{\alpha \in C_p} 0 + \sum_{\alpha \in C_p} A(pK,0,1)\big) \\
	= & p(p+1)A(pK,0,1) .
	\end{align*}
	
	Hence, we get $\tensor*[_p]{\mu}{_2} = p(p+1)\lambda_p$ with $\lambda_{p}$ given in \eqref{lambda p} as required. The same exact argument also proves that $\tensor*[_p]{\mu}{_4} = p(p+1)\lambda_p$.
	
	To show $\tensor*[_p]{\mu}{_3} = p^2\lambda_{p}^{2} + p^3 + p$, we use condition 2(c) of Propositon \ref{prop3.1} and get  
	\begin{align*}
	(K_p h_3 K_p \cdot F)_{\beta} = & \Big(p^2 A(p^{-1}\beta ) + p^2A(p\beta) + p \sum_{(\alpha_1,\alpha_2) \in C_p \times C_p} A(\alpha_1^{-1} \beta \alpha_2)\Big)\\
	= & p^2\cdot 0 + p^2A(p^2K,0,p)+p((p+1)A(K,0,1)) \\
	= & p^2A(p^2K,0,p)+(p^2+p)A(K,0,1)\\
	= & p^2(pA(K,0,1) + A(p^2K,0,1))\\
	+ & p(p+1)A(K,0,1) \numberthis \label{step5.3}\\
	= & p^3A(K,0,1) + pA(K,0,1) \\
	+ & p^2(A(p^2K,0,1) + A(K,0,1)).
	\end{align*}
	The $A(p^{-1}\beta )=0$ since $p\nmid \beta$. By Lemma \ref{l51}, for each $\alpha_2$ there is a unique $\alpha_1$ such that $\alpha_1 ^{-1}\beta \alpha_2 \in \mathcal{O}$. Hence, there are total $(p+1)$ copies of $A(K,0,1)$, one for each $\alpha_2$. 
	%Should I explain this line more? This has a couple steps which I don't think are completely obvious.	
	For \eqref{step5.3}, we are using our recurrence relation \eqref{2b} of Definition \ref{rec} to expand $A(p^2K,0,p) = pA(K,0,1) + A(p^2K,0,1)$. 
	
	It now suffices to prove $A(p^2K,0,1) + A(K,0,1) = \lambda_p ^2 A(K,0,1)$ to show $\tensor*[_p]{\mu}{_3} = p^2\lambda_{p}^{2} + p^3 + p$ as in \eqref{3.5}. However, we know $\lambda_p ^2 A(K,0,1) = \lambda_p (\lambda_p A(K,0,1)) = \lambda_p (A(pK,0,1))$ from the argument for $\tensor*[_p]{\mu}{_2}$. Let $\beta'$ be such that $A(\beta') = A(pK,0,1)$. Then, it follows that 
	\begin{align*}
	p(p+1)\lambda_p A(pK,0,1) = & (K_p h_2 K_p \cdot F)_{\beta'} \\ 
	= &  p\big( \sum_{\alpha \in C_p} A(\beta' \bar{\alpha}^{-1}) + \sum_{\alpha \in C_p} A(\bar{\alpha}\beta')\big)\\
	= & p(A(K,0,1) + A(p^2K,0,p) \\
	  & + pA(p^2K,0,1))\hfill \numberthis \label{step5.4}\\
	= & p(A(K,0,1) + pA(K,0,1)+ A(p^2K,0,1) \\
	  & + pA(p^2K,0,1)) \hfill \numberthis \label{step5.5}\\
	= & p(p+1)(A(K,0,1)+A(p^2K,0,1)).
	\end{align*}
	We use Lemma \ref{l51} to expand out the sums to obtain \eqref{step5.4}. Since $p|pK$, there exists a unique $\alpha \in C_p$ such that $\alpha^{-1}\beta' \in \mathcal{O}$ and $A(\beta' \bar{\alpha}^{-1})\\ = A(K,0,1)$ for that $\alpha$. $A(\beta' \bar{\alpha}^{-1})= 0$ in the other $p$ cases. In the second sum, there exists a unique $\alpha \in C_p$ such that $A(\bar{\alpha}\beta') = A(p^2K,0,p)$. In the other $p$ cases, $A(\bar{\alpha}\beta') = A(p^2K,0,1)$. We use the recurrence relation \eqref{2b} of Definition \ref{rec} again to obtain \eqref{step5.5}. Hence,  $\tensor*[_p]{\mu}{_3} = p^2\lambda_p+p^3+p$ as required, completing the first case.
	
	\textbf{Case 2: }$K = p^nK_0$ with $n>0$. Let $\beta$ be such that $A(\beta) = A(p^nK_0,0,1)$ where $K_0$ is an even number co-prime to $p$.
	\begin{align*}
	(K_p h_2 K_p \cdot F)_{\beta} = & p\big( \sum_{\alpha \in C_p} A(\beta \bar{\alpha}^{-1}) + \sum_{\alpha \in C_p} A(\bar{\alpha}\beta)\big)\\
	= & p(A(p^{n-1}K_0,0,1) + A(p^{n+1}K_0,0,p)\\ 
	&+ pA(p^{n+1}K_0,0,1)) \\
	%= & p(A(p^{n-1}K_0,0,1) + pA(p^{n+1}k/p^2,0,1) + A(p^{n+1}k/,0,1) \\ &+ pA(p^{n+1}K_0,0,1)) \\
	= & p(A(p^{n-1}K_0,0,1) + A(p^{n+1}K_0,0,1) \\
	& +  pA(p^{n-1}K_0,0,1)+ pA(p^{n+1}K_0,0,1)) \numberthis \label{step5.6}\\
	= & p(p+1)(A(p^{n-1}K_0,0,1) + A(p^{n+1}K_0,0,1)) .
	\end{align*}
	We use Lemma \ref{l51} again to write the sums in terms of $A(K,u,n)$. As $p|p^{n}K_0$ but $p\nmid \beta$, there exists a unique $\alpha \in C_p$ such that $\beta \alpha^{-1} \in \mathcal{O}$, for which $A(\beta \bar{\alpha}^{-1}) = A(p^{n-1}K_0,0,1)$. As before, $A(\beta \bar{\alpha}^{-1}) = 0$ in all the other $p$ cases. In the second sum, there exists a unique $\alpha\in C_p$ such that $A(\bar{\alpha}\beta) = A(p^{n+1}K_0,0,p)$. In the other $p$ cases, $A(\bar{\alpha}\beta) = A(p^{n+1}K_0,0,1)$. We obtain equation \eqref{step5.6} then by using the recurrence relation \eqref{2b} of Definition \ref{rec} to expand $A(p^{n+1}K_0,0,p)$. Hence, we get $\tensor*[_p]{\mu}{_2} = p(p+1)\lambda_p$ with $\lambda_{p}$ given in \eqref{lambda p} as required. Once again, the same exact argument also proves that $\tensor*[_p]{\mu}{_4} = p(p+1)\lambda_p$.
	
	To show that $\tensor*[_p]{\mu}{_3} = \lambda_{p}^2p^2 +p^3+p$, we have to consider two subcases: $n=1$ and $n \geq 2$. We will set both cases up and prove them together.
	\textbf{Subcase 1:} Letting $n=1$, we get 
	\begin{align*}
	(K_p h_3 K_p \cdot F)_{\beta} = & \Big(p^2 A(p^{-1}\beta ) + p^2A(p\beta) + p \sum_{(\alpha_1,\alpha_2) \in C_p \times C_p} A(\alpha_1^{-1} \beta \alpha_2)\Big)\\
	= & p^2\cdot 0  + p^2A(p^3K_0,0,p)+p((p+1)A(pK_0,0,1)\\
	+ & pA(pK_0,0,1)) \\
	%= & p^2(\sum_{n|d}pnA(p^3k/p^2n^2,0,1)+ \sum_{n|d}nA(p^3k/n^2,0,1))+(2p^2+p)A(pK_0,0,d)\\
	= & p^2(pA(pK_0,0,1) + A(p^3K_0,0,1))\\
	+ & (2p^2+p)A(pK_0,0,1) \numberthis \label{step5.7}\\
	= & p^3A(pK_0,0,1) + pA(pK_0,0,1) + p^2(A(p^3K_0,0,1) \\
	+ & 2A(pK_0,0,1)).
	\end{align*}
	We will again use Lemma \ref{l51} to simplify the terms in the summation. The first term, $ A(p^{-1}\beta ) =0$ as $p\nmid \beta$. Since $p|pk$, there exist unique $\alpha'_2 \in C_p$ such that $p|\beta \alpha'_2$. For that $\alpha'_2$, $A(\alpha_1^{-1}\beta \alpha'_2)= A(pK_0,0,1)$ for every $\alpha_1 \in C_p$. In the other $p$ cases of $\alpha_2$'s, there exists a unique $\alpha_1 \in C_p$ such that $\alpha_1 ^{-1 }\beta \alpha_2 \in \mathcal{O}$ and $A(\alpha_1^{-1}\beta \alpha'_2)= A(pK_0,0,1)$. We use the recurrence relation \eqref{2b} of Definition \ref{rec} to expand $A(p^3K_0,0,p)$ to obtain \eqref{step5.7}. \\
	\textbf{Subcase 2:}  $n \geq 2$
	\begin{align*}
	(K_p h_3 K_p \cdot F)&_{\beta}\\  =& \Big(p^2 A(p^{-1}\beta ) + p^2A(p\beta) + p \sum_{(\alpha_1,\alpha_2) \in C_p \times C_p} A(\alpha_1^{-1} \beta \alpha_2)\Big)\\
	= & p^2\cdot0 + p^2A(p^{n+2}K_0,0,p)+p(pA(p^nK_0,0,1)\\ 
	+ & A(p^nK_0,0,p) + pA(p^nK_0,0,1)) \\
	= & p^2(pA(p^nK_0,0,1) + A(p^{n+2}K_0,0,1)) \\
	+ & p((2p+1)A(p^nK_0,0,1) +  pA(p^{n-2}K_0,0,1))\numberthis \label{step5.8} \\
	= & p^2(A(p^{n+2}K_0,0,1) + A(p^{n-2}K_0,0,1) + 2A(p^nK_0,0,1))\\ 
	+ & (p^3+p)A(p^nK_0,0,1).
	\end{align*}
	Once again, using Lemma \ref{l51} to simplify the summation, we get $A(p^{-1}\beta )=0$ as $p\nmid \beta$. Since $p|p^nK_0$, there exists a unique $\alpha'_2 \in C_p$ such that $p|\beta \alpha'_2$. Since now $p\mid \beta \alpha'_2$, there exists a unique $\alpha_1 \in C_p$ such that $(\alpha_1^{-1}\beta \alpha'_2) =A(p^nK_0,0,p)$. In other $p$ cases of $\alpha_1$, $A(\alpha_1^{-1}\beta \alpha'_2) = A(p^nK_0,0,1)$. In the other $p$ cases of $\alpha_2$'s there exist unique $\alpha_1$'s such that $\alpha_1 ^{-1 }\beta \alpha_2 \in \mathcal{O}$ and $A(\alpha_1^{-1}\beta \alpha'_2) = A(p^nK_0,0,1)$. We use recurrence relation \eqref{2b} twice, more precisely, once for the $p^2A(p^{n+2}K_0,0,p)$ and once for $A(p^nK_0,0,p)$ to obtain \eqref{step5.8}. 
	
	To prove $\tensor*[_p]{\mu}{_3} = p^2\lambda_p+p^3+p$ in both subcase 1 and subcase 2, it suffices to show that $A(p^{n+2}K_0,0,1) + A(p^{n-2}K_0,0,1) + 2A(p^nK_0,0,1) = \lambda_p^2 A(p^nK_0,0,1)$ where $A(p^{n-2}K_0,0,1) = 0$ for the first subcase. For this, we use the identity $\lambda_p\big(A(p^{n+1}K_0,0,1) + A(p^{n-1}K_0,0,1)\big) = \lambda_p^2 A(p^nK_0,0,1)$ from before. All that is left to show now is that $A(p^{n+2}K_0,0,1) + A(p^{n}K_0,0,1) = \lambda_p A(p^{n+1}K_0,0,1)$ and $A(p^{n}K_0,0,1) + A(p^{n-2}K_0,0,1) = \lambda_p A(p^{n-1}K_0,0,1)$. Both of these are easy to prove and follow from the computation of $(K_p h_2 K_p \cdot F)_\beta$ done at the start of \textbf{Case 2}.
	
	Thus, the Hecke eigenvalues $\tensor*[_p]{\mu}{_1}, \tensor*[_p]{\mu}{_2}, \tensor*[_p]{\mu}{_3}, \tensor*[_p]{\mu}{_4}$ for $F$ satisfy the equation (\ref{3.5}) as required. Rest of the proof follows from the proof of Proposition 6.2 in \cite{MNP}.
\end{proof}

Proposition \ref{prop5.1} gives us the exact structure of $\Pi_{F,p}$ for all odd primes $p$. Next we give a description of $\Pi_{F,2}$ and $\Pi_{F,\infty}$.

\begin{prop} \label{Prop5.2}
	a) The local component $\Pi_{F,2}$ is the unique irreducible constituent of the unramified principal series representation $\text{Ind}_{\mathcal{B}_2(B_2)}^{\GL_2(B_2)}(\chi_1 \times \chi_2)$ with $\chi_1 ,\chi_2$ unramified characters of $B_2^\times$ such that
	$$
	\chi_1(\varpi_2) = -\sqrt{2}\epsilon, \chi_2(\varpi_2) = -1/\sqrt{2}\epsilon.
	$$
	
	b) At the prime $p= \infty$, the archimedean component $\Pi_{F,\infty}$ is isomorphic to the principal series representation $\text{Ind}^{\GL_2(\mathbb{H})}_{\mathcal{B}_2(\HH)}(\chi_{\pm \frac{\sqrt{-1}r}{2}})$ where
	$$\chi_s\left(\begin{bmatrix}
	a & *\\
	0 & d
	\end{bmatrix} \right) = \nu(ad^{-1})^s$$
\end{prop}

\begin{proof}[Proof of Proposition \ref{Prop5.2}]
	a) For the structure of $\Pi_{F,2}$, it is again enough to show that the Hecke eigenvalues $_2\mu_1$, $\tensor*[_2]{\mu}{_2}$ satisfy the equation (\ref{3.6}). The proof of this is simpler than the odd prime case. From the Maass space condition \eqref{2a} in Definition \ref{rec}, we have 
	$$A(2K,u+1,n) = \frac{-3\epsilon}{\sqrt{2}}A(K,u,n) - A(\frac{K}{2},u-1,n)$$
	which gives us
	$$2(A(2K,u+1,n) + A(K/2,u-1,n)) = -3\sqrt{2}\epsilon A(K,u,n).$$
	Let $\beta \in \Ss$ such that $A(\beta) = A(K,u,n)$. Then, in terms of $\beta$, the above condition can be written as
	$$2(A(\beta \varpi_2) + A(\beta \varpi_2^{-1})) = -3\sqrt{2}\epsilon A(\beta).$$
	Comparing with condition 1 of Proposition 3.1, we get that the Hecke eigenvalue $\tensor*[_2]{\mu}{_2} = -3\sqrt{2}\epsilon$. Rest of the argument follows from Sections 5.2 and 6.1 of \cite{MNP}.
	
	b) The proof for the structure of $\Pi_{F,\infty}$ is the same as in Section 6.1 of \cite{MNP} since we still have a Maass form with Casimir eigenvalue $-\frac{1}{2}\left(\frac{r^2}{4} +1\right)$.
\end{proof}

\section{Jacquet Langlands correspondence}
Let $B_\A$ denote the adelization of $B$ with $B_p = B \otimes_\Q \Q_p$ as before. 
Badulescu and Renard in Theorem 1.4 of \cite{4} give a map $\textbf{G}$ from the automorphic representations on $\GL_2(B_\A)$ to those on $\GL_4(\A)$. Let $D\GL_2(B_\A)$ and $D\GL_4(\A)$ denote the discrete series representations of $\GL_2(B_\A)$ and $\GL_4(\A)$ respectively. For $G'= \GL_2(B_2)$ or $\GL_2(\HH)$ and $G = \GL_4(\Q_2)$ or $\GL_4(\R)$ respectively, we denote by $\mathcal{C}(G')$ the category of smooth representations of $G'$ (in the non-archimedean case) or the category of Harish-Chandra modules (in the archimedean case) with a fixed maximal compact subgroup $K$ of $G'$. Let $\mathcal{R}(G')$ denote the Grothendieck group of the category of finite length representations in $\mathcal{C}(G')$. If $g \in \GL_n(B_p)$ in the non-archimedean case for some n or $g \in \GL_n(\HH)$ for the archimedean case, we say that $g$ is regular semisimple if the characteristic polynomial of $g$ has distinct roots in an algebraic closure of $\Q_p$ or $\C$ respectively. If $\pi \in \mathcal{R}(G)$, then we denote by $\Theta_\pi$ the function character of $\pi$ as a locally constant map, stable under conjugation, defined on the set of regular semisimple elements of $G$. We say that $g' \in G$ corresponds to $g \in \GL_4(\Q_2)$ or $\GL_4(\R)$ respectively if $g$ and $g'$ are regular semisimple and have the same characteristic polynomial and we write $g \leftrightarrow g'$. For a unitary irreducible smooth representation $u$ of $G$, we say that $u$ is compatible if there is a unique unitary smooth irreducible representation $u'$ of $G'$ such that $\Theta_u(g) = \varepsilon(u)\Theta_{u'}(g')$ for any $g \leftrightarrow g'$, where $\varepsilon(u) \in \{-1,1\}$. We denote the map $u \rightarrow u'$ by $|\textbf{LJ}_v|$ where $v = 2$ or $\infty$. We say a discrete series $\pi$ of $\GL_4(\A)$ is $B-compatible$ if $\pi_v$ is compatible at both $v=2$ and $v=\infty$.

\begin{thm}[Theorem 1.4 of \cite{4}]

\begin{enumerate}[label=\alph*)] 
\item There is a unique map\\ $\textbf{G}: D\GL_2(B_\A) \rightarrow D\GL_4(\A)$ such that for every $\pi' \in D\GL_2(B_\A)$, if $\pi = \textbf{G}(\pi')$, then one has: 
\begin{itemize}
\item $\pi$ is $B$-compatible
\item if $v \neq 2,\infty$, then $\pi_v$ = $\pi'_v$
\item if $v = 2$ or $v = \infty$, then $|\textbf{LJ}_v|(\pi_v) = \pi'_v$
\end{itemize}
The map $\textbf{G}$ is injective. The image of $\textbf{G}$ is the set of all $B$-compatible discrete series of $\GL_4(\A)$.

\item If $\pi' \in D\GL_2(B_\A)$, then the multiplicity of $\pi'$ in the discrete spectrum is one (multiplicity one theorem).

\item If $\pi', \pi'' \in D\GL_2(B_\A)$ and $\pi'_v \simeq \pi''_v$ for almost all $v$, then $\pi' = \pi''$ (strong multiplicity one theorem)
	\end{enumerate}
\end{thm}


For a representation $\pi$ of $\GL_n$, we will say $\pi = \MW(\sigma,k)$ if a discrete series representation $\pi$ is the unique irreducible quotient of the induced representation $\nu^{(k-1)/2}\sigma \times \nu^{(k-3)/2}\sigma \times \ldots \times \nu^{-(k-1)/2}\sigma$. Here $\sigma$ is cuspidal and $\nu$ is the global character given by product of local characters i.e. absolute value of reduced norm. For $\GL_4$, the possible values of $k$ will be 1,2 or 4. In each of these cases, $\sigma$ will be a cuspidal representation of $\GL_4$, $\GL_2$ and $\GL_1$ respectively over the appropriate group.\\
Let $\Pi$ denote the image of $\Pi_F$ under $\textbf{G}$. 
We will use following results from Proposition 18.2 of \cite{4} to find conditions on $\Pi$.

\begin{prop}[Proposition 18.2 of \cite{4}]
Let $\pi = \MW(\rho,k)$ be a representation of $\GL_4(\A)$. 
\begin{enumerate}[label=\alph*)]
\item There exists $k_\rho \in \{1,2\}$ such that $\pi$ is $B$-compatible if and only if $k_\rho \mid k$.

\item Let $\pi'$ be a discrete series of $\GL_2(B_\A)$ and let $\pi = \textbf{G}(\pi')$. Then $\pi'$ is cuspidal if and only if $\pi$ is of the form $\MW(\rho,k_\rho)$. 
\end{enumerate}
\end{prop}

By Proposition 18.2 part b) of \cite{4}, since $\Pi_F$ is cuspidal, its image $\Pi$ is of the form $\MW(\sigma,k_\sigma)$. By Proposition 18.2 part a) of \cite{4}, $k_\sigma \mid d$ when the dimension of the division algebra is $d^2$. In our case, the division algebra is a quaternion algebra, so $d=2$. Hence, $k_\sigma \mid 2$ implying $k_\sigma = 2$ or $k_\sigma = 1$. The latter condition is same as $\sigma$ being cuspidal.

\begin{prop} \label{prop5.3}
	Let $F \in \Ms$ be a cuspidal Hecke eigenform with $\Pi_F$ the associated representation of $\GL_2(B_\A)$. Then $\textbf{G}(\Pi_F) = \MW(\sigma, 2)$ for some cuspidal representation  $\sigma$ of $\GL_2(\A)$. 
\end{prop}
\begin{proof}
	We will show that $ \textbf{G}(\Pi_F) = \Pi$ is not cuspidal, which is equivalent to showing $k_\sigma \neq 1$. Since $k_\sigma = 1$ or $k_\sigma =2$, this proves the proposition.
	
	For the sake of contradiction, assume $k_\sigma =1$. Therefore, $\Pi$ is a cuspidal automorphic representation of $\GL_4(\A)$. Then, by equation (14) of Sarnak \cite{7}, for every odd prime $p$ we have 
	$$\left|\log_p\left(\left|\alpha_i\left(\Pi_p\right)\right|_p\right)\right| \leq \frac{1}{2} - \frac{1}{4^2 +1}.$$
	Here $\alpha_i(\Pi_p)$ denotes the i-th Satake parameter of $\Pi$ at prime $p$, $| \mid_p$ denotes the p-adic valuation and the outer $| \mid$ denotes the standard absolute value. We have $\alpha_i(\Pi_p) = \chi_{i}(p)$ with $\chi_{i}(p)$ as given in \eqref{5.1}. This, in particular, tells us that 
	$$\left|\log_p \left( \left|p^{1/2} \frac{\lambda_p \pm \sqrt{\lambda_p ^2 - 4}}{2}\right|_p \right)\right| \leq \frac{1}{2} - \frac{1}{4^2 +1}$$ and 
	$$\left|\log_p \left( \left|p^{-1/2} \frac{\lambda_p \pm \sqrt{\lambda_p ^2 - 4}}{2}\right|_p \right)\right| \leq \frac{1}{2} - \frac{1}{4^2 +1}.$$
	Therefore, we can write
	\begin{equation} 
	\left|\log_p \left(\left|p^{\pm 1/2}\right|_p\right) + \log_p \left( \left| \frac{\lambda_p \pm \sqrt{\lambda_p ^2 - 4}}{2}\right|_p \right)\right| \leq \frac{1}{2} - \frac{1}{17}.\label{1/2-}
	\end{equation}
	
	Let $\log_p \left( \left| \frac{\lambda_p + \sqrt{\lambda_p ^2 - 4}}{2} \right|_p \right) = \alpha^+$ and $\log_p \left(\left|\frac{\lambda_p - \sqrt{\lambda_p ^2 - 4}}{2} \right|_p\right) = \alpha^-$ for convenience of notation. Note that $\alpha^+ + \alpha^- = 0$. Then \eqref{1/2-} implies that
	\begin{align*}
	\left|\frac{1}{2} + \alpha^+ \right| \leq \frac{1}{2} - \frac{1}{17} &&
	\left|\frac{1}{2} + \alpha^- \right| \leq \frac{1}{2} - \frac{1}{17} \\
	\left|\frac{-1}{2} + \alpha^+ \right| \leq \frac{1}{2} - \frac{1}{17} &&
	\left|\frac{-1}{2} + \alpha^- \right| \leq \frac{1}{2} - \frac{1}{17}
	\end{align*}
	In particular, we get that $$-\frac{1}{2} + \frac{1}{17} \leq \frac{1}{2} + \alpha^+  \leq \frac{1}{2} - \frac{1}{17} \quad \text{ and } \quad-\frac{1}{2} + \frac{1}{17} \leq \frac{-1}{2} + \alpha^+ \leq \frac{1}{2} - \frac{1}{17}.$$ 
	Simplifying, we get
	$$ -1 + \frac{1}{17} \leq  \alpha^+  \leq - \frac{1}{17} \text{\quad and \quad } \frac{1}{17} \leq  \alpha^+  \leq 1 - \frac{1}{17}$$
	both of which cannot be simultaneously true. This gives us a contradiction to our starting assumption that $k_\sigma = 1$. Hence, $k_\sigma \neq 1$ which implies $k_\sigma =2 $ as required.
\end{proof}

From Proposition \ref{prop5.3} we obtain an irreducible cuspidal automorphic representation $\sigma$ of $\GL_2(\A)$. We will next describe the local components of $\sigma$ and use that to construct $f \in \SG$ which can lift to $F$.

\section{Description of $\sigma$}

Let $\sigma$ be as from Proposition \ref{prop5.3}, with $\sigma \simeq \otimes_{p \leq \infty} \sigma_p$. 
For an odd prime $p$, let $\chi_{p}$ be the unramified character of $\Q_p^\times$ such that $\chi_{p} (p) = \frac{\lambda_p + \sqrt{\lambda_p ^2 - 4}}{2}$ for $\lambda_{p}$ as in Proposition \ref{prop5.1}. At the prime $p= \infty$, let $\chi_{\infty}(a) = |a|^s$ where $s = \frac{\sqrt{-1}r}{2}$. For the prime $p=2$, let $\chi$ be an unramified character of $\Q_2^\times$ with $\chi(2) = - \epsilon$ for $\epsilon$ as in condition \eqref{2a} of Definition \ref{rec}.

\begin{prop}\label{Prop5.4}
	Let $\sigma \simeq \otimes_{p \leq \infty} \sigma_p$ be the irreducible cuspidal automorphic representation of $\GL_2(\A)$ from Proposition \ref{prop5.3}. Then
	\begin{equation} \label{5.4}
	\sigma_p = 
	\begin{cases}
	\ind_{\mathcal{B}_2(\Q_p)}^{\GL_2(\Q_p)}(\chi_{p}\times \chi_{p}^{-1}) \quad &\text{ for odd } p< \infty, \\
	\chi St_{\GL_2} \quad &\text{ for } p = 2, \\
	\ind_{\Bg(\R)}^{\GL_2(\R)}(\chi_\infty \times \chi_\infty^{-1}) \quad &\text{ for } p = \infty.
	\end{cases}
	\end{equation}
\end{prop}

\begin{proof}
We will use the local Jacquet-Langlands map $$\textbf{C}: \GL_2(B_p) \rightarrow \GL_4(\Q_p)$$ to explicitly write down $\sigma_p$ at each prime $p$. We will use the same notation for the map from $\GL_2(\HH)$ to $\GL_4(\R)$. The groups $\GL_2(B_p)$ and $\GL_4(\Q_p)$ are isomorphic for every odd prime $p$, hence following section 1.5 of \cite{4} the maps is identity for odd prime $p$. For $p=2$, the map is described in Theorem 3.2 of \cite{5} and in Section 1.3 of \cite{4} for $p= \infty$. 

Since the $\textbf{C}$ map is identity at every odd prime $p$, we have $\Pi_{F,p} = \Pi_p$ where $\Pi_p$ is the local component of $\Pi$ at prime $p$. Let $P_{2,2}$ denote the 2,2-parabolic subgroup of $\GL_4$. From Proposition \ref{prop5.3} we know that $\Pi_{F,p} = \MW(\sigma_p,2)$. We also know that $\Pi_{F,p}$ is the spherical constituent of $\ind^{\GL_4(\Q_p)}_{\BG(\Q_p)}(\chi_1 \times \chi_2 \times\chi_3 \times\chi_4)$. We will denote $\ind^{\GL_4(\Q_p)}_{\BG(\Q_p)}(\chi_1 \times \chi_2 \times\chi_3 \times\chi_4)$ by $\ind_{\BG(\Q_p)}^{\GL_4(\Q_p)}(\chi')$ for ease of notation. The reduced norm here is just $\nu = |\det|$.

If $\Pi_{F,p} = \MW(\sigma_p,2)$, then such a $\sigma_p$ is unique (see \cite{13} Section 8). Hence, to show the structure of $\sigma_p$, it is enough to prove the following claim:
\begin{claim}
	For every odd prime $p$, $$\ind^{\GL_4(\Q_p)}_{\BG(\Q_p)}(\chi_1 \times \chi_2 \times\chi_3 \times\chi_4) \simeq \ind^{\GL_4(\Q_p)}_{P_{2,2}(\Q_p)}(\nu^{1/2}\sigma_p \times \nu^{-1/2} \sigma_p)$$ 
	for 
	$\sigma_p = \ind_{\Bg(\Q_p)}^{\GL_2(\Q_p)}(\chi_p \times \chi_p ^{-1})$.
\end{claim}

	Using method similar to 6.5 in \cite{3}, define the map 
	$$L:\ind_{P_{2,2}(\Q_p)} ^{\GL_4(\Q_p)}( \nu^{1/2}\sigma_p \times \nu^{-1/2} \sigma_p) \rightarrow \ind_{\BG(\Q_p)}^{\GL_4(\Q_p)}(\chi')$$
	by 
	$$(Lh)(g) := (h(g))(I_2 , I_2).$$
	
	Here $h$ is a function in $\ind_{P_{2,2}(\Q_p)} ^{\GL_4(\Q_p)}( \nu^{1/2}\sigma_p \times \nu^{-1/2}\sigma_p)$ and $I_n$ is the identity matrix in $\GL_n(\Q_p)$. 
	We have to show this map is well defined and is an isomorphism.
	
	To show that $L$ is well-defined we have to prove that for any $A \in \BG$, $Lh$ satisfies $(Lh)(Ag) = \delta_{\BG}^{1/2}(A)\chi'(A)(Lh)(g)$. For $A = \begin{bmatrix}
	a & * & * & * \\
	0 & b & * & * \\
	0 & 0 & c & * \\
	0 & 0 & 0 & d \\
	\end{bmatrix}$ \\
	we have
	\begin{align*}
	\delta_{\BG}^{1/2}(A)&\chi'(A)(Lh(g))=\\ &=|a^3bc^{-1}d^{-3}|^{1/2}\chi_1(a)\chi_2(b)\chi_3(c)\chi_4(d)(Lh(g))\\
	&= |a|^{3/2}|b|^{1/2}|c|^{-1/2}|d|^{-3/2}|a|^{1/2}\chi_p(a)|b|^{1/2}\chi_p ^{-1}(b)\\ &\quad |c|^{-1/2}\chi_p(c)|d|^{-1/2}\chi_p^{-1}(d)(Lh(g))\\
	&=|a|^2 |b| |c|^{-1} |d|^{-2} \chi_{p}(a)\chi_{p}^{-1}(b)\chi_{p}(c)\chi_{p}^{-1}(d)(Lh(g))
	\end{align*}
	
	On the other hand
	\begin{align*}
	(L&h)(Ag) = \\	&=\left|\det \begin{bmatrix}
	a & * \\
	0 & b \\
	\end{bmatrix}\right| \left|\det \begin{bmatrix}
	c & * \\
	0 & d \\
	\end{bmatrix}\right| ^{-1}
	 \nu^{1/2}\left(\begin{bmatrix}
	a & * \\
	0 & b \\
	\end{bmatrix}\right)
	 \nu^{-1/2}\left(\begin{bmatrix}
	c & * \\
	0 & d \\
	\end{bmatrix}\right)\\
	& \left( \left(\sigma_p
	\left(\begin{bmatrix}
	a & * \\
	0 & b \\
	\end{bmatrix}\right) , \sigma_p\left(\begin{bmatrix}
	c & * \\
	0 & d \\
	\end{bmatrix}\right)\right)h(g)\right)(I_2 , I_2)\\
	&= |a||b||c|^{-1}|d|^{-1}|a|^{1/2}|b|^{1/2} |c|^{-1/2}|d|^{-1/2}h(g)\left(\begin{bmatrix}
	a & * \\
	0 & b \\
	\end{bmatrix} \times \begin{bmatrix}
	c & * \\
	0 & d \\
	\end{bmatrix}\right)\\
	&= |a||b||c|^{-1}|d|^{-1}|a|^{1/2}|b|^{1/2}|c|^{-1/2}|d|^{-1/2} \\ & \quad |a|^{1/2}|b|^{-1/2}\chi_p(a)\chi_p ^{-1}(b)|c|^{1/2}|d|^{-1/2}\chi_p(c)\chi_p^{-1}(d)h(g)(I_2 , I_2 )\\
	&=|a|^2 |b| |c|^{-1} |d|^{-2} \chi_{p}(a)\chi_{p}^{-1}(b)\chi_{p}(c)\chi_{p}^{-1}(d)h(g)(I_2 , I_2)\\
	&=|a|^2 |b| |c|^{-1} |d|^{-2} \chi_{p}(a)\chi_{p}^{-1}(b)\chi_{p}(c)\chi_{p}^{-1}(d)(Lh(g))\\
	&=\delta_{\BG}^{1/2}(A)\chi'(A)(Lh(g))
	\end{align*}
	
	To prove injectivity, we look at two functions $h_1$ and $h_2$ in\\ $\ind_{P_{2,2}(\Q_p)} ^{\GL_4(\Q_p)}( \nu^{1/2}\sigma_p \times \nu^{-1/2}\sigma_p)$. 
	By definition, then $Lh_1 = Lh_2$ implies $h_1(g)(I_2, I_2) = h_2(g)(I_2 ,  I_2)$ for every $g$. Applying $( (\nu^{1/2}\sigma_p)(s_1) \times (\nu^{-1/2}\sigma_p)(s_2))$ for $s_1,s_2 \in \GL_2(\Q_p)$ to both sides, we get $h_1(g)(s_1 , s_2) = h_2(g)(s_1 , s_2)$. 
	Therefore, $h_1 = h_2$. 
	
	To show that it is an isomorphism, we construct an inverse map $$\tilde{L}:\ind_{\BG(\Q_p)}^{\GL_4(\Q_p)}(\chi')\rightarrow \ind_{P_{2,2}(\Q_p)} ^{\GL_4(\Q_p)}( \nu^{1/2}\sigma_p \times \nu^{-1/2} \sigma_p) $$ given by $$(\tilde{L}h)(g)(b_1 , b_2) =  h\left(\begin{bmatrix}
	b_1 & 0 \\
	0 &  b_2\\
	\end{bmatrix}g\right)$$ for $b_1,b_2 \in \mathcal{B}_2(\Q_p)$. We can verify that it is well defined by similar computation as above and it is easy to see that $L\circ \tilde{L}$ is identity. Hence $L$ is an isomorphism of representation.

\begin{claim}
	At prime $p =\infty$,  
	$$|\textbf{C}|(\Pi_{F,\infty}) \simeq \ind^{\GL_4(\R)}_{P_{2,2}(\R)}(\nu^{1/2}\sigma_\infty \times \nu^{-1/2} \sigma_\infty)$$ for $\sigma_\infty = \ind_{\Bg(\R)}^{\GL_2(\R)}(\chi_\infty \times \chi_\infty ^{-1})$ with $\chi_\infty (a) = |a|^s$ and $s = \frac{\sqrt{-1}r}{2}$. 
\end{claim}

	For $p=\infty$, note that calculations in Section 6 of \cite{MNP} for the description of $\Pi_\infty$ are for a general element $F \in \MG$ and are independent of any lifting properties. 
	Hence, $\Pi_{F,\infty}$ is the irreducible component of $\ind_{\Bg(\HH)}^{\GL_2(\HH)}( \chi'_s \times \chi_s ^{\prime-1})$ with $\chi'_s = \nu ^{\prime s} (x)$ and $s = \frac{\sqrt{-1}r}{2}$. 
	Here $\nu'$ denotes the reduced norm of $\mathbb{H}$ at infinity and is equal to the square root of the absolute value. 
	
	Following Section 1.3 from \cite{5}, the image of the Jacquet-Langlands correspondence is $\ind_{P_{2,2}(\R)}^{\GL_4(\R)}(\xi_s \times \xi_s^{-1})$ where $\xi_s = |\textbf{C}|(\chi'_s) $ and $\xi_s ^{-1} = |\textbf{C}|(\chi_s^{\prime -1})$. 
	Here $\xi_s, \xi_s ^{-1}$ are characters of $\GL_2(\R)$ with $\xi_s = \chi_s \circ \det$ and $\chi_s (a) = |a|^s$ for $s = \frac{\sqrt{-1}r}{2}$. 
	Therefore, $\Pi_\infty = \ind_{P_{2,2}(\R)}^{\GL_4(\R)}(\xi_s \times \xi_s^{-1}).$ 
	Now, $\ind_{P_{2,2}(\R)}^{\GL_4(\R)}(\xi_s \times \xi_s^{-1})$ is the irreducible quotient of $\ind_{P_{2,2}(\R)}^{\GL_4(\R)}(\tau_s \times \tau_{-s})$ where $\tau_s = \ind_{\Bg(\R)}^{\GL_2(\R)}(| \mid^{1/2}\chi_s \times |\mid^{-1/2}\chi_s)$. Hence, we obtain the following isomorphism
	\begin{align*}
	\ind_{P_{2,2}(\R)}^{\GL_4(\R)}&(\tau_s \times \tau_{-s}) \\ 
	&\simeq \ind_{\BG(\R)}^{\GL_4(\R)}(|\mid^{1/2}\chi_s\times |\mid^{-1/2}\chi_s\times |\mid^{1/2}\chi_{-s} \times |\mid^{-1/2}\chi_{-s}) \\
	&\simeq \ind_{\BG(\R)}^{\GL_4(\R)}(|\mid^{1/2}\chi_s\times |\mid^{1/2}\chi_{-s}\times |\mid^{-1/2}\chi_{s} \times |\mid^{-1/2}\chi_{-s}) \\
	& \simeq \ind_{P_{2,2}(\R)}^{\GL_4(\R)}(\nu^{1/2}\sigma_s \times \nu^{-1/2}\sigma_s)
	\end{align*}
	where $\sigma_s = \ind_{\Bg(\R)}^{\GL_2(\R)}(\chi_s \times \chi_{-s})$ and $\nu = |\det|$.
	
	However, we know from global Jaquet Langlands of Section 5.2 that $\Pi_\infty$ is also of the form $MW(\sigma_\infty,2)$ which is the irreducible quotient of $\ind_{P_{2,2}(\R)}^{\GL_4(\R)}(\nu^{1/2}\sigma_\infty \times \nu^{-1/2}\sigma_\infty)$. Hence, by uniqueness, we get $\sigma_\infty = \sigma_s$ as in the claim.
	
	Structure of $\sigma_p$ for $p$ odd and $p = \infty$ cases is proved by the claims above. We will now show case $p=2$.
	
	Let $\rho$ and $\rho'$ be unitary representations of $\GL_2(\Q_2)$ and $B_2^\times$ respectively such that $\textbf{C}(\rho') = \rho$.
	At prime $p=2$, Theorem 3.2 from \cite{4} tells us $\textbf{C}(\overline{u}(\rho',2)) = u(\rho,2)$ where
	$$u(\rho,k) = Lg(\Pi_{i=0}^{k-1}\nu^{(k-1)/2 - i}\rho), \quad \overline{u}(\rho',k) = Lg(\Pi_{i=0}^{k-1}\nu'^{(k-1)/2 - i}\rho')$$
	with $Lg$ denoting the unique irreducible quotient. Here, $\nu = |\det|$ and $\nu'$ is the reduced norm. 
	In our case, we have $u(\rho,2) = \Pi_2$ and $\overline{u}(\rho',2) = \Pi_{F,2}$.
	We know $\Pi_{F,2}$, unlike at odd primes $p$, is a representation of $\GL_2(B_2)$. On the other hand have $\Pi_2 =\MW(\sigma_2,2)$, hence $k = 2$. Therefore,  $\Pi_{F,2} =  \text{Ind}_{\mathcal{B}_2(\Q_p)}^{\GL_2(B_2)}(\chi_1 \times \chi_2) = \MW(\rho',2)$. Hence, $\rho'$ is a one dimensional representation of $B^\times$ given by a character $\chi' = \chi \circ \nu'$ for an unramified character $\chi$ of $\Q_2^\times$. Comparing with Proposition 5.2, we get $\chi'(\varpi_2) = -\epsilon$. According to Section 56 of \cite{11} such a character corresponds to twisted Steinberg representation $\chi St$ of $\GL_2(\Q_2)$. Hence, $\sigma_2 = \chi St$ with $\chi(2) = - \epsilon $.
	
	
\end{proof}

\chapter{Main Theorem}

\section{Distinguished vector in $\sigma$}
Theorem \ref{Th1} showed the `necessary' condition that if a given $F \in \MG$ is a lift then $F \in \mathcal{M}^* (\GL_2(\mathcal{O}),r)$. We wish to prove a converse of Theorem \ref{Th1}, i.e. the `sufficient' condition. We will first show this under the extra hypothesis that $F$ is a Hecke eigenform and in the last section prove it in all generality for all $F \in \Ms$.

\begin{thm}\label{Thm5.1}
	Let $F \in \mathcal{M}^*(\GL_2(\mathcal{O}),r)$ such that $F$ is a cuspidal Hecke eigenform. Then,  there is a $f \in \SG$, a Hecke eigenform, such that $F = F_f$.
\end{thm}

From Section 5.3, we know that $\sigma_p$ is unramified principal series at every prime $p \neq 2$. Hence, the new vector at every prime $p\neq \{2,\infty\}$ is the unique spherical vector $\psi_p$ stable under $K_p = \GL_2(\Z_p)$. At $p= \infty$, we have the unique weight zero fixed vector $\psi_\infty$ which is stable under $K_\infty = O_2(\R)$. 

At $p=2$, the representation is an unramified twist of Steinberg and hence the conductor is $p$. 
Therefore, the new vector $\psi_2$ is invariant under $$K_2 = \left\{\begin{bmatrix}
a & b \\
c & d \\
\end{bmatrix}\Big| a,d\in \Z_2^\times, b\in \Z_2, c\in 2\Z_2 \right\}.$$

Let $\psi = \otimes_{p\leq \infty}\psi_p \in V_\sigma$.
It satisfies 
$$ \psi(z\gamma g k) = \psi(g) \quad  \text{ for } \gamma \in \GL_2(\Q), z \in Z(\GL_2(\A)), k \in \Pi_{p \leq \infty} K_p.$$

For $g_\infty = \begin{bmatrix}
a & b\\
c & d
\end{bmatrix} \in \GL_2(\R)$, let $g_\infty(i) = \frac{ai+b}{ci+d} = \tau \in \mathfrak{h}$. Consider the function $f_\psi : \mathfrak{h} \rightarrow \C$ associated to $\psi$ defined as $f_\psi(\tau) = f_\psi(g_\infty(i)) = \psi(g_\infty \otimes_{p<\infty} 1_p)$ where $1_p$ is the identity of $\GL_2(\Q_p)$. Then, for $\gamma \in \Gamma_0(2)$ we have
\begin{align*}
f_\psi(\gamma(\tau)) &= \psi((\gamma g_\infty)\otimes_{p<\infty}1_p)\\
&= \psi((\otimes_{p \leq \infty}\gamma^{-1})((\gamma g_\infty)\otimes_{p<\infty}1_p)) \hfill &\because \otimes_{p \leq \infty} \gamma^{-1} \in \GL_2(\Q)\\
&=\psi(g_\infty \otimes_{p<\infty}\gamma^{-1})\\
&=\psi((g_\infty \otimes_{p<\infty}1_p)k) \hfill & k= (1\otimes_{p<\infty}\gamma^{-1})\\
&=\psi(g_\infty \otimes_{p<\infty}1_p)  &\because k\in \Pi_{p \leq \infty}K_p\\
&=f_\psi(\tau)
\end{align*}

Hence $f_\psi$ is invariant under the action of $\Gamma_0(2)$. 
Since the local representation $\sigma_\infty$ at $p=\infty$ associated with the vector $\psi_\infty$  is principal series, the function $f_\psi$ is a Maass form.

Following Lemma 9 from \cite{14} for $n=1$, the map $\psi \rightarrow f_\psi$ is Hecke equivariant. The structure of $\sigma_p$ from Section 5.3 allows us to find the Hecke eigenvalues for $f_\psi$ at all odd prime $p < \infty$. Following Proposition 3.1.2 of \cite{12}, the function $f_\psi$ is an eigenfunction of the Atkin-Lehner involution with eigenvalue $-\chi(2) = \epsilon$ from \eqref{5.4} and Hecke eigenvalue $\lambda_2 = \chi(2) = -\epsilon$. By Proposition 4.6.6 of \cite{2}, the Hecke eigenvalue for odd primes $p$ with $\sigma_p = \ind_{\Bg(\Q_p)}^{\GL_2(\Q_p)}(\chi_p \times \chi_p ^{-1})$ would be $(\chi_{p}(p) + \chi_{p}^{-1}(p)) = \lambda_p$. Note that we are using the action of the Hecke algebra as in (30) of \cite{14} here, hence the lack of $p^{1/2}$.

The eigenvalue for the hyperbolic Laplacian is obtained from the Hecke eigenvalue at infinity as by Proposition 2.5.4 from \cite{2}. Following the notation from Bump \cite{2}, in this case, $s_1 = \frac{\sqrt{-1}r}{2}$ and $s_2 = -\frac{\sqrt{-1}r}{2}$. Hence, $s = \frac{1}{2}(\sqrt{-1}r + 1) = \frac{1+ \sqrt{-1}r}{2} $. Then, the eigenvalue for the Laplacian is given by $s(1-s) = \left(\frac{1}{4} + \frac{r^2}{4}\right)$. Hence, $f_\psi$ belongs to $\SG$ as required. We will do an additional verification that this $f_\psi$ does indeed lift to $F$. This will complete the proof of Theorem \ref{Thm5.1}.


\section{Fourier coefficients of $f_\psi$}
Let $N = 4^a b$, where $a,b$ are non-negative integers and $4 \nmid b $. For $F\in \Ms$ we can define a sequence of numbers
\begin{equation}
c(-N) \coloneqq \frac{A(2N,u,1)}{\sqrt{2N}} + \epsilon \frac{A(N,u-1,1)}{\sqrt{N}} \label{CN}
\end{equation}
where \[ 
u = 
\begin{dcases}
2a & \text{if }b \equiv 1,3 \mod (4) \\
2a+1 & \text{if }b \equiv 2 \mod (4) \\ 
\end{dcases}
\]
It can be proved that this sequence of numbers $\{c(-N)\}$ satisfy \eqref{3.3} in terms of $K,u$ and $n$ just by reversing the argument in proof of Theorem \ref{Th1}.
\begin{align*}
A(K,u,&n)= \sum_{d \mid n} d A(K/d^2,u,1)
\\=& \sqrt{K} \sum_{d \mid n} \frac{A(K/d^2,u,1)}{\sqrt{K/d^2}}
\\=&\sqrt{K} \sum_{d \mid n} \Bigg(\sum_{t=0}^u \Bigg( (-\epsilon)^t	 \frac{A((K/d^2)/2^{t},u-t,1)}{\sqrt{(K/d^2)/2^{t}}} \\
&- (-\epsilon)^{t+1} \frac{A((K/d^2)/2^{t+1},u-(t+1),1)}{\sqrt{(K/d^2)/2^{t+1}}}\Bigg)\Bigg)
\\=& \sqrt{K} \sum_{t=0}^u \sum_{d \mid n} (-\epsilon)^t \Bigg(\frac{A(K/(2^{t}d^2),u-t,1)}{\sqrt{K/(2^{t}d^2)}}\\ 
&+ \epsilon \frac{A(K/(2^{t+1}d^2),u-t-1,1)}{\sqrt{K/(2^{t+1}d^2)}}\Bigg)\\ 
=&\sqrt{K} \sum_{t=0}^u \sum_{d \mid n} (-\epsilon)^t c\left(-\frac{K}{2^{t+1}d^2}\right) 
\end{align*}
We will show that $f_\psi \in \SG$ lifts to $F$ by showing that these $\{c(N)\}$ are the coefficients of $f$ for all $N<0.$ 
This is to say, if $c_\psi(N)$ are the Fourier coefficients of $f_\psi$, then $c(-N) = c_\psi(-N)$ for all $N>0$. 

For every odd prime $p$, since $f_\psi$ is a Hecke eigenform, $c_\psi(N)$ satisfy 
\begin{equation}
p^\frac{1}{2} c_\psi(pN) + p^{\frac{-1}{2}} c_\psi(N/p)  = \lambda_p c_\psi(N). \label{5.5}
\end{equation}
by equation (5.7) of \cite{MNP}, with $c_\psi(N/p) = 0$ if $p\nmid N$. Rewriting, we get
\begin{equation}
c_\psi(pN) = p^{-1/2}\lambda_p c_\psi(N) - p^{-1} c_\psi(N/p). \label{5.6}
\end{equation}
Since $f$ is a Hecke eigenform at prime $p=2$, from equation (5.6) of \cite{MNP}, its Fourier coefficients also satisfy 
\begin{equation}
c_\psi(2N) = -\frac{\epsilon}{2}c_\psi(N).	\label{5.7}
\end{equation}

Equations \eqref{5.6} and \eqref{5.7} together allows us to write $c_\psi(-N)$ in terms of $c_\psi(-1)$, $\lambda_p$ and $\epsilon$ for all $N$. This also shows that $c_\psi(-1)$ is in fact non-zero. 

\begin{lem}
	The sequence of numbers $\{c(N)\}$ as defined in \eqref{CN} satisfy equations \eqref{5.5} and \eqref{5.7}
\end{lem}

\begin{proof}
	To prove $\{c(N)\}$ satisfy \eqref{5.5}, we will use that 
	$$\lambda_p = \frac{A(p^{n+1}K_0,u,1) + A(p^{n-1}K_0,u,1)}{ A(p^nK_0,u,1)}.$$
	Note that this statement is slightly different than our claim in \eqref{lambda p} since we no longer assume $u=0$. This statement is still true however, since the Hecke computation from Proposition 3.1 (b) holds true for any general $A(K,u,n)$ and not just with $A(K,0,1)$ as we used for Proposition \ref{prop5.1}. It can also be calculated explicitly via the same argument as in proof of Proposition \ref{prop5.1}. In the computation below, we assume $c(-N/p)=0$ and $A(N/p,u-1,1)=0$ are $0$ if $p\nmid N$.
	\begin{align*}
	p^\frac{1}{2} c(-pN)& + p^\frac{-1}{2} c(-N/p) \\= &  \left(p^{1/2}\frac{A(2pN,u,1)}{\sqrt{2pN}} + \epsilon p^{1/2} \frac{A(pN,u-1,1)}{\sqrt{pN}} \right) \\
	+ &  \left( p^{-1/2}\frac{A(2N/p,u,1)}{\sqrt{2N/p}} + \epsilon p^{-1/2} \frac{A(N/p,u-1,1)}{\sqrt{N/p}}\right)\\
	=& \frac{A(2pN,u,1)}{\sqrt{2N}} + \epsilon \frac{A(pN,u-1,1)}{\sqrt{N}} \\  +& \frac{A(2N/p,u,1)}{\sqrt{2N}} + \epsilon \frac{A(N/p,u-1,1)}{\sqrt{N}}\\
	=& \left( \frac{A(2pN,u,1) + A(2N/p,u,1)}{\sqrt{2N}} \right) \\ 
	 +& \left( \epsilon \frac{A(pN,u-1,1)+A(N/p,u-1,1) }{\sqrt{N}} \right)\\
	=&  \frac{\lambda_p A(2N,u,1)}{\sqrt{2N}} + \epsilon \frac{\lambda_p A(N,u-1,1)}{\sqrt{N}}\\
	=& \lambda_p \Big(\frac{A(2N,u,1)}{\sqrt{2N}} + \epsilon \frac{A(N,u-1,1)}{\sqrt{N}}\Big)\\
	=& \lambda_p c(-N)
	\end{align*}
	Thus, $\{c(-N)\}$ satisfy \eqref{5.5}. To show equation \eqref{5.7}, we use the condition \eqref{2a} from Definition \ref{rec}. From $c(-N)$ as in \eqref{CN}, we get
	\begin{align*}
	c(-2N) =& \frac{A(4N,u+1,1)}{\sqrt{4N}} + \epsilon \frac{A(2N,u,1)}{\sqrt{2N}}\\
	=& \left(\frac{-3\epsilon}{\sqrt{2}}\frac{A(2N,u,1)}{\sqrt{4N}} - \frac{A(4N,u-1,1)}{\sqrt{4N}}\right) + \epsilon\frac{A(2N,u,1)}{\sqrt{2N}}\\
	=& \frac{-3\epsilon}{2}\frac{A(2N,u,1)}{\sqrt{2N}} + \epsilon\frac{A(2N,u,1)}{\sqrt{2N}} - \frac{A(N,u-1,1)}{2\sqrt{N}}\\
	=& \frac{-\epsilon}{2}\frac{A(2N,u,1)}{\sqrt{2N}} - \frac{1}{2}\frac{A(N,u-1,1)}{\sqrt{N}}\\
	=& \frac{-\epsilon}{2}c(-N)
	\end{align*}
	
\end{proof}
Since $c(N)$ satisfy \eqref{5.5} and \eqref{5.7}, $c(-1)$ is also not $0$. Then, we can normalize the Fourier coefficients $c_\psi(-N)$ so that $c_\psi(-1)= c(-1)$. Since both $\{c(-N)\}$ and $\{c_\psi(-N)\}$ satisfy \eqref{5.5} and \eqref{5.7}, this implies $c(-N) = c_\psi(-N)$ for all $N$. Therefore, the Fourier coefficients of $f_\psi$ satisfy \eqref{3.3} and hence, it is a Hecke eigenform in $\SG$ whose lift $F_f =F$, completing the proof of Theorem \ref{Thm5.1}.

\section{Main result for non-Hecke eigenforms}
We would like to generalize the result of Theorem $\ref{Thm5.1}$ to all $F \in \Ms$. We will do so by proving that $\MG$ has a Hecke eigenbasis and showing that the Maass space $\Ms$ is stable under the action of all the Hecke operators given in Proposition $\ref{prop3.1}$. If $\Ms$ is stable, then it has a Hecke eigenbasis $\{F_i\}$ which are lifts of some $\{f_i\}$ for $f_i \in \SG$ by Theorem \ref{Thm5.1}. Then by linearity of the defining condition $\eqref{3.3}$, $F = \sum_i a_i F_i$ would be a lift of $\sum_i a_i f_i \in \SG$. Let $\Gamma \subset \GL_2(\mathcal{O})$ be a subgroup of finite index. For Maass forms $F,G$ over $\mathcal{M}(\Gamma,r)$ with one of them cuspidal, we can define their Petersson inner product by 
\begin{equation}
\langle F,G \rangle = \frac{1}{Vol(\Gamma\backslash\GL_2(\mathcal{O}))} \int_{\Gamma\backslash \GL_2(\HH)/Z^+K}F(g)\overline{G(g)} dg \label{Pet}
\end{equation}
where the Haar measure $dg$ is given by $\frac{dxdy}{y^2}$ when $g = \begin{bmatrix}
y & x \\
0 & 1 \\
\end{bmatrix}$ as in Section 2.1.

\begin{prop}
	$\MG$ has a basis of forms that are simultaneous eigenvectors of the Hecke algebra $\otimes \mathcal{H}(G_p,K_p)$ and the subspace of cusp forms has an orthogonal basis of Hecke eigenfunctions with respect to the Petersson inner product on the 5-dimensional hyperbolic as in \eqref{Pet}
	
	
\end{prop}
\begin{proof}
	The Hecke algebra acting on $\MG$ is $\otimes \mathcal{H}(G_p,K_p)$ as in Section 5.2 of \cite{MNP}. By Theorem 6 from Section 8 of \cite{Sat}, the algebra $\mathcal{H}(G_p,K_p)$ is commutative for every prime $p$. 
	
	By Theorem 1 on page 8 of \cite{HC}, dim$(\MG) < \infty$. This means the space $\MG$ is a finite dimensional vector space with a commutative algebra $\otimes \mathcal{H}(G_p,K_p)$ of operators acting on it. The final step in the proof is to show that the operators commute with their adjoint with respect to the Petersson inner product.
	
	Let $g$ be such that $K_pgK_p$ is one of the generators of the Hecke algebra $\mathcal{H}(G_p,K_p)$ which according to Proposition \ref{prop3.1} are $h_1,h_2,h_3$ and $h_4$ for odd $p$ and 
	$\begin{bmatrix}
	\varpi_2 & 0\\
	0 & \varpi_2
	\end{bmatrix}$ and $\begin{bmatrix}
	\varpi_2 & 0\\
	0 & 1
	\end{bmatrix}$ for $p=2$. Let $K_pgK_p = \sqcup_i K_pg_i = \sqcup_i g_iK_p$. Then $K_pg^{-1}K_p = \sqcup_i K_p g_i^{-1}$ and $K_pg^{-1}zK_p = \sqcup_i K_p g_i^{-1}z$ for $z\in Z$ an element of the center. Let $F,G$ be cusp forms in $\MG$. To find the adjoint operator under the Petersson inner product of \eqref{Pet}, we can look at
	
	\begin{align*}
	\langle T(g)F,G\rangle = \sum_i \langle F|_{g_i},G\rangle = & \sum_i \langle F|_{g_i},G|_z\rangle \\
	= & \sum_i \langle F|_{g_i},G|_z|_{g_i^{-1}}|_{g_i}\rangle\\
	= & \sum_i \langle F,G|_{zg_i^{-1}}\rangle\\
	= & \sum_i \langle F,T(zg^{-1})G\rangle.
	\end{align*}
	Hence, to show that the Hecke operators commute with their adjoint, it suffices to show that $T(zg^{-1})$ is a generator up to a element of the center. Note that while $F,G$ are cusp forms with respect to $\GL_2(\mathcal{O})$, the forms $G|_{zg^{-1}}$ might be modular only with respect to some smaller subgroup $\Gamma$ hence hence we define Petersson inner product for general subgroup $\Gamma$ rather than just $\GL_2(\mathcal{O})$.  
	
	Now, taking $z = \begin{bmatrix}
	\varpi_2 & 0\\
	0 & \varpi_2
	\end{bmatrix}$ and the Weyl group element $w = \begin{bmatrix}
	0 & 1\\
	1 & 0
	\end{bmatrix} \in K_2$, we can see that $K_2gK_2 = K_2wzg^{-1}wK_2$. Similarly, taking $z = \begin{bsmallmatrix}
	p & & & \\
	& p & & \\
	& & p & \\
	& & & p 
	\end{bsmallmatrix}$ and $w = \begin{bsmallmatrix}
	& & & 1\\
	& & 1 & \\
	& 1 & & \\
	1 & & & 
	\end{bsmallmatrix} \in K_p$, we can show that $K_ph_2K_p = K_pwzh_4^{-1}wK_p$, $K_ph_3K_p = K_pwzh_3^{-1}wK_p$ and $K_ph_4K_p = K_pwzh_2^{-1}wK_p$. Letting $T_{p,i}$ denote the Hecke operator of $K_ph_iK_p$, this shows 
	$$T^*_{2,1}=T_{2,1}, \quad T^*_{2,2}=T_{2,2}, $$ and
	$$T^*_{p,1}=T_{p,1}, \quad T^*_{p,2}=T_{p,4}, \quad T^*_{p,3}=T_{p,3}, \quad T^*_{p,4}=T_{p,2}$$
	for every prime odd $p>2$. 
\end{proof}

\begin{thm} \label{MainTh}
	The following are equivalent.
	\begin{enumerate}
		\item $F$ is a lift from an Atkin-Lehner eigenform $f \in \SG$ with eigenvalue $\epsilon \in \{\pm 1 \}$ and which is a Hecke eigenform at $p=2$.
		\item $F$ is an element of the space $\Ms$
	\end{enumerate}
\end{thm}

\begin{proof}
	As mentioned before, it is enough to show that $\Ms$ is stable under the action of the Hecke Algebra. We will prove that for any Hecke operator $T_{p,i}$ and any $F \in \Ms$, the image of the action $T_{p,i}F \in \Ms$ by showing $T_{p,i}F$ satisfies all the conditions of Definition \ref{rec}. 
	
	Condition 1 of Definition \ref{rec} follows from the fact that we can write the coefficients $A'(\beta)$ of $T_{p,i}F$ in terms of $A(K,u,n)$ using Proposition \ref{prop3.1} by case by case decomposition similar to proof of Proposition \ref{prop5.1}.
	We showed in proof of Proposition \ref{Prop5.2} that condition $\eqref{2a}$ is actually equivalent to $F$ being a Hecke eigenform at prime $p=2$. Hence, $T_{2,2}F = -(3\sqrt{2}\epsilon) F$ for all $F \in \Ms$. 
	Checking the recurrence relations for $T_{p,i}F$ where $p$ is an odd prime requires computation. We will show this case by case with $A(\beta) = A(p^mK,u,p^ln)$ where $p\nmid Kn$. Our cases will be for $T_{p,2}F$ and $T_{p,3}F$ with $m=2l$ and $m>2l$. The computation for $T_{p,4}$ is identical to $T_{p,2}$ and hence will not be shown separately.

To simplify computation, we will use a simpler version of our recurrence relation.
	\begin{align*}A(p^mK,u,p^ln)=& \sum_{d\mid p^ln}dA\left(\frac{p^mK}{d^2},u,1\right) \\
	=& \sum_{i=0}^{l} \sum_{d'\mid n} p^i d' A \left(\frac{p^mK}{p^{2i} d'^2},u,1\right)\numberthis\label{step}\\
	=& \sum_{i=0}^{l} p^i \sum_{d'\mid n} d' A\left(\frac{p^{m-2i}K}{d'^2},u,1\right)\\
	=& \sum_{i=0}^{l}p^iA(p^{m-2i}K,u,n). \numberthis\label{nrec}
	\end{align*}
	To obtain \eqref{step}, we wrote $d = p^i d'$ and split the sum over $d\mid p^ln$ into sum over $0\leq i \leq l$ and $d'\mid d$.
	
	For ease of notation, we will refer to the Fourier coefficients of the $T_{p,i}F$ in terms of $K,u$ and $n$ as $T_{p,i}F(K,u,n)$. 
	In terms of recurrence relation \eqref{nrec}, the result we want to show will be
	$$T_{p,i}F(p^mK,u,p^ln) = \sum_{i=0}^{l}p^iT_{p,i}F(p^{m-2i}K,u,n).$$
	 
	\textbf{Case 1}: We will start with computation for $T_{p,2}F$ and $m=2l$. By convention any term $A(p^mK,u,p^ln)=0$ if either $l$ or $m$ is negative. In this case, $p^l$ exactly divides $p^mK$ so $\beta = p^l\beta_0$ with $p\nmid|\beta_0|^2$. Therefore, by Lemma \ref{l51}, we have
	\begin{align*}
	T_{p,2}F(p^mK,u,&p^ln) = \\ =& (p+1)A(p^{m+1}K,u,p^ln)  + (p+1)A(p^{m-1}K,u,p^{l-1}n)\\
	= & (p+1)(A(p^{m+1}K,u,p^ln) + A(p^{m-1}K,u,p^{l-1}n))
	\end{align*}
	and 
	\begin{align*}
	T_{p,2}F(p^mK,u,n) = & pA(p^{m+1}K,u,n) + A(p^{m+1}K,u,pn)\\ +& A(p^{m-1}K,u,n)\\
	=&pA(p^{m+1}K,u,n) + pA(p^{m-1}K,u,n) \\
	+& A(p^{m+1}K,u,n) + A(p^{m-1}K,u,n)\\
	= & (p+1)(A(p^{m+1}K,u,n) + A(p^{m-1}K,u,n)).
	\end{align*}
	Note, we are using our recurrence relation from step 2 to step 3 to expand the terms with $pn$. 
	Hence,
	\begin{align*}
	\sum_{i=0}^l p^i T_{p,2}&F(p^{m-2i}K,u,n) =\\ =& \sum_{i=0}^{l-1} p^i T_{p,2}F(p^{m-2i}K,u,n) + p^l T_{p,2}F(K,u,n)\\ 
%	=& \sum_{i=0}^{l-1}  p^i (pA(p^{m+1-2i}K,u,n) + A(p^{m+1-2i}K,u,pn) \\
%	+& A(p^{m-1-2i}K,u,n)) + p^l(p+1)A(pK,u,n)\\
%	=&\sum_{i=0}^{l-1} p^i ( pA(p^{m+1-2i}K,u,n) + pA(p^{m-1-2i}K,u,n) \\
%	 +& A(p^{m+1-2i}K,u,n) + A(p^{m-1-2i}K,u,n)) +p^l(p+1)A(pK,u,n)\\
	=& \sum_{i=0}^{l-1}p^i (p+1)(A(p^{m+1-2i}K,u,n) +  A(p^{m-1-2i}K,u,n))\\
	&+ p^l(p+1)(A(pK,u,n)+0)
	\end{align*}


	We compare that with
	\begin{align*}
	T_{p,2}F(&p^mK,u,p^ln) = (p+1)(A(p^{m+1}K,u,p^ln) + A(p^{m-1}K,u,p^{l-1}n)) \\ 
	=& (p+1)\left( \sum_{i=0}^{l}p^i A(p^{m+1-2i}K,u,n) + \sum_{i=0}^{l-1} p^i A(p^{m-1-2i}K,u,n)\right)\\
	=& (p+1)\left( \sum_{i=0}^{l-1}p^i A(p^{m+1-2i}K,u,n) + \sum_{i=0}^{l-1} p^i A(p^{m-1-2i}K,u,n)\right)\\
	 &+ p^l(p+1)A(pK,u,n)
	\end{align*}
	to obtain the desired result.

\textbf{Case 2}: Now, consider the computation for $T_{p,2}F$ with $m>2l$. 
	The expansion for $T_{p,2}F(p^{m}K,u,n)$ from \textbf{Case 1} is still valid here. Hence,
	\begin{align*}
	\sum_{i=0}^l p^i T_{p,2}&F(p^{m-2i}K,u,n) =\\ =& \sum_{i=0}^{l}p^i (p+1)(A(p^{m+1-2i}K,u,n) +  A(p^{m-1-2i}K,u,n))
	\end{align*}
	In this case, $\beta = p^l \beta_0$ but $p \mid |\beta_0|^2$. Therefore, by Lemma \ref{l51}, we have
	\begin{align*}
	T_{p,2}F&(p^mK,u,p^ln) =\\ 
	=& pA(p^{m+1}K,u,p^ln) +A(p^{m+1}K,u,p^{l+1}n) \\
	+& pA(p^{m-1}K,u,p^{l-1}n) +A(p^{m-1}K,u,p^ln)\\ 
	=& p \sum_{i=0}^{l}p^i A(p^{m+1-2i}K,u,n) + \sum_{i=0}^{l+1} p^i A(p^{m+1-2i}K,u,n)\\
	+&p \sum_{i=0}^{l-1}p^i A(p^{m-1-2i}K,u,n) + \sum_{i=0}^{l} p^i A(p^{m-1-2i}K,u,n)\\
	=& (p+1) \sum_{i=0}^{l}p^i A(p^{m+1-2i}K,u,n) + p^{l+1} A(p^{m-1-2l}K,u,n)\\
	+&(p+1) \sum_{i=0}^{l-1}p^i A(p^{m-1-2i}K,u,n) + p^l A(p^{m-1-2l}K,u,n)\\
	=& (p+1) \sum_{i=0}^{l}p^i A(p^{m+1-2i}K,u,n) + (p+1)p^{l} A(p^{m-1-2l}K,u,n)\\
	+&(p+1) \sum_{i=0}^{l-1}p^i A(p^{m-1-2i}K,u,n) \\
	=& (p+1)\left( \sum_{i=0}^{l}p^i A(p^{m+1-2i}K,u,n) + \sum_{i=0}^{l-1} p^i A(p^{m-1-2i}K,u,n)\right).
	\end{align*}
	Comparing the two sums we have the equality.
	
	\textbf{Case 3}: Now we move on to the computation for $T_{p,3}F$, starting with $m=2l$. Once again we have $p^l$ exactly dividing $p^mK$ so $\beta = p^l\beta_0$ with $p\nmid|\beta_0|^2$. Therefore, by Lemma \ref{l51}, we have
	\begin{align*}
	T_{p,3}F(p^m&K,u,p^ln) =\\
	= & p^2A(p^{m-2}K,u,p^{l-1}n)+p^2A(p^{m+2}K,u,p^{l+1}n)\\
	+ & p(p(p+1)A(p^mK,u,p^{l-1}n)+(p+1)A(p^mK,u,p^ln))\\
	= & p^2A(p^{m-2}K,u,p^{l-1}n)+p^2A(p^{m+2}K,u,p^{l+1}n)\\
	+ & (p^2+p)(pA(p^mK,u,p^{l-1}n) + A(p^mK,u,p^ln))
	\end{align*}
	and 
\begin{align*}
T_{p,3}F(p^m&K,u,n) = \\
= & p^2(0)+p^2A(p^{m+2}K,u,pn) \\
+ & p(pA(p^mK,u,n)+pA(p^mK,u,n)+A(p^mK,u,pn))\\
= & p^2A(p^{m+2}K,u,pn)+2p^2A(p^mK,u,n)+pA(p^mK,u,pn)\\
= & p^2(pA(p^mK,u,n) +A(p^{m+2}K,u,n)) + 2p^2A(p^mK,u,n)\\
+ & p(pA(p^{m-2}K,u,n) + A(p^mK,u,n))\\
= & p^2A(p^{m+2}K,u,n) + p^2A(p^{m-2}K,u,n)\\
+ & (p^3+2p^2+p) A(p^mK,u,n).
\end{align*}
Hence,
\begin{align*}
\sum_{i=0}^l p^i T_{p,3}&F(p^{m-2i}K,u,n) =\\
%=& \sum_{i=0}^l p^i (p^2A(p^{m+2-2i}K,u,pn) + 2p^2A(p^{m-2i}K,u,n) + pA(p^{m-2i}K,u,pn))\\
%=&\sum_{i=0}^{l} p^i(p^2(A(p^{m+2-2i}K,u,n)+pA(p^{m-2i}K,u,n)) + 2p^2A(p^{m-2i}K,u,n) \\ 
%+&p(A(p^{m-2i}K,u,n)+pA(p^{m-2-2i}K,u,n)))\\
= & \sum_{i=0}^{l}p^i(p^2A(p^{m-2-2i}K,u,n) +p^2A(p^{m+2-2i}K,u,n) \\ 
+ & (p^3+2p^2+p)A(p^{m-2i}K,u,n))\\
= & \sum_{i=0}^{l-1}p^i(p^2A(p^{m-2-2i}K,u,n) +p^2A(p^{m+2-2i}K,u,n) \\ 
+ & (p^3+2p^2+p)A(p^{m-2i}K,u,n))\\
+ & p^l(p^2A(p^2K,u,n) + (p^3+2p^2+p)A(K,u,n))\\
\end{align*}
We separated the term of $i=l$ since it expands differently for the case of $p^{m-2-2i}$. 

Comparing it with
\begin{align*}
T_{p,3}F(p^mK&,u,p^ln)=  \\
= & p^2A(p^{m-2}K,u,p^{l-1}n)+p^2A(p^{m+2}K,u,p^{l+1}n)\\
+ &  (p^2+p)(pA(p^mK,u,p^{l-1}n) + A(p^mK,u,p^ln))\\ 
= & p^2 \sum_{i=0}^{l-1}p^iA(p^{m-2-2i}K,u,n)+p^2 \sum_{i=0}^{l+1}p^iA(p^{m+2-2i}K,u,n)\\
+ & (p^2+p)\left(p\sum_{i=0}^{l-1}p^iA(p^{m-2i}K,u,n)+\sum_{i=0}^{l}p^iA(p^{m-2i}K,u,n)\right)\\
= & p^2 \sum_{i=0}^{l-1}p^iA(p^{m-2-2i}K,u,n)+p^2\sum_{i=0}^{l-1}p^iA(p^{m+2-2i}K,u,n) \\ 
+ & p^2 p^lA(p^2K,u,n) + p^2 p^{l+1} A(K,u,n)\\ 
+ & (p^2+p)\left(p\sum_{i=0}^{l-1}p^iA(p^{m-2i}K,u,n)+\sum_{i=0}^{l}p^iA(p^{m-2i}K,u,n)\right)\\
+ & (p^2+p) p^l A(K,u,n)\\  
= & \sum_{i=0}^{l-1}p^i(p^2A(p^{m-2-2i}K,u,n) +p^2A(p^{m+2-2i}K,u,n) \\ 
+ & (p^3+2p^2+p)A(p^{m-2i}K,u,n))\\
+ & p^l(p^2A(p^2K,u,n) + (p^3+2p^2+p)A(K,u,n))
\end{align*}
we get that the two sums are equal as required.

	
	
	\textbf{Case 4}: Finally, let $m>2l$ with $A(p^{m-2-2l}K,u,n) =0$ if $m=2l+1$. Then, we have
	
	\begin{align*}
	\sum_{i=0}^l p^i &T_{p,3}F(p^{m-2i}K,u,n) =\\ 
	=& \sum_{i=0}^l p^i (p^2A(p^{m+2-2i}K,u,pn) + 2p^2A(p^{m-2i}K,u,n)\\  
	+& pA(p^{m-2i}K,u,pn))\\
	=&\sum_{i=0}^{l} p^i(p^2(A(p^{m+2-2i}K,u,n)+pA(p^{m-2i}K,u,n))\\
	+& 2p^2A(p^{m-2i}K,u,n) +p(A(p^{m-2i}K,u,n)+pA(p^{m-2-2i}K,u,n)))\\
	=& \sum_{i=0}^{l}p^i(p^2A(p^{m-2-2i}K,u,n) +p^2A(p^{m+2-2i}K,u,n) \\
	+& (p^3+2p^2+p)A(p^{m-2i}K,u,n)
	\end{align*}
	Note, we are using our recurrence relation from step 1 to step 2 to expand the terms with $pn$. 
	
	We compare that with
	\begin{align*}
	T_{p,3}F(p^mK,u,&p^ln) = p^2A(p^{m-2}K,u,p^{l-1}n)\\
	+& p^2A(p^{m+2}K,u,p^{l+1}n) +p(p^2A(p^mK,u,p^{l-1}n)\\
	+ & 2pA(p^mK,u,p^ln) + A(p^mK,u,p^{l+1}n))\\
	= & p^2 \sum_{i=0}^{l-1}p^iA(p^{m-2-2i}K,u,n)+p^2\sum_{i=0}^{l+1}p^iA(p^{m+2-2i}K,u,n)\\
	+& p^3\sum_{i=0}^{l-1}p^iA(p^{m-2i}K,u,n)+2p^2\sum_{i=0}^{l}p^iA(p^mK,u,n)\\
	+& p\sum_{i=0}^{l+1}p^iA(p^mK,u,n)\\
	= & p^2 \sum_{i=0}^{l-1}p^iA(p^{m-2-2i}K,u,n)+p^2\sum_{i=0}^{l}p^iA(p^{m+2-2i}K,u,n) \\ +&p^2p^{l+1}A(p^{m-2l}K,u,n) +p^3\sum_{i=0}^{l-1}p^iA(p^{m-2i}K,u,n)\\ +&2p^2\sum_{i=0}^{l}p^iA(p^mK,u,n)+ p\sum_{i=0}^{l}p^iA(p^mK,u,n)\\  
	+& pp^{l+1}A(p^{m-2-2l}K,u,n).
	\end{align*}
	
	Rearranging the sum, we get 
	\begin{align*}
	T_{p,3}F(p^mK,u,p^ln)&= \sum_{i=0}^{l}p^i(p^2A(p^{m-2-2i}K,u,n)\\ 
	& +\sum_{i=0}^{l}p^ip^2A(p^{m+2-2i}K,u,n) \\ 
	&+ \sum_{i=0}^{l}p^i(p^3+2p^2+p)A(p^{m-2i}K,u,n)\\
	&=\sum_{i=0}^l p^i T_{p,3}F(p^{m-2i}K,u,n) 
	\end{align*}
	as required.
\end{proof}

\appendix
\chapter{Converse theorem for $\Gamma_0(4)$}

\section{Converse Theorem}
Following the method of construction of Muto, Narita and Pitale in \cite{MNP}, a possible approach to the problem is via a proper converse theorem. If we could use the two recurrence relations from Definition \ref{rec} and Proposition \ref{prop1}, we can perhaps infer about the analytic properties of $C(-N)$ from $A(\beta)$. The following proposition fulfills the role of the required converse theorem for the case $N=4$.

\begin{proposition}
	If a Maass form $f\in \SG$ has a Fourier expansion $\sum_{-\infty}^{\infty}a_n \sqrt{y} K_{\nu}(2\pi |n|y) e^{2\pi i nx}$ with $a_0 = 0$ and if 
	$$\Lambda (s,f) := N^{(\frac{s-1/2}{2})} \pi^{(-s+ \epsilon)} \Gamma\Big(\frac{s+\epsilon + \nu}{2}\Big) \Gamma\Big(\frac{s+\epsilon - \nu}{2}\Big) \sum \frac{a_n}{n^s}$$ 
	satisfies the functional equation $\Lambda(s,f) = (-1)^\epsilon \Lambda(1-s,f)$ with $\epsilon = 0$ if $a_n = a_{-n}$ and $\epsilon = 1$ if $a_n = -a_{-n}$ for $N \leq 4$, then $f$ is a Maass form over $\Gamma_0(N)$.
\end{proposition}

We will use Lemma 1.9.2 from \cite{2} to prove this proposition.

\begin{lemma}[\textbf{1.9.2}] %1.9.2
	If $f$ is an eigenvector for the Laplace operator, then $f(iy) = \frac{\partial f(iy)}{\partial x} = 0$ for all $y > 0$ $\Rightarrow$ $f(z) = 0$ for all $z$. 
\end{lemma}

\begin{proof}
 Action of $\gamma$ doesn't change Laplace invariance and leaves the eigenvalue unchanged. Hence, $h(z)$ is also a Laplace eigenfunction and we can use the above lemma to it.  From the existence of Fourier expansion, we know that $f(z)$ is invariant under 
[$\begin{smallmatrix}
1 & 1 \\
0 & 1 \\
\end{smallmatrix}]$. Since these two matrices generate $\Gamma_0(N)$ for $N\leq 4$, we would have shown $f(z)$ is $\Gamma_0(N)$ invariant.Now [$\begin{smallmatrix}
0 & -1 \\
N & 0 \\
\end{smallmatrix} ][
\begin{smallmatrix}
1 & -1 \\
0 & 1 \\
\end{smallmatrix}]
[\begin{smallmatrix}
0 & -1 \\
N & 0 \\
\end{smallmatrix}]^{-1}
=     
[\begin{smallmatrix}
1 & 0 \\
N & 1 \\
\end{smallmatrix}]$. So showing that $f(iy) = \pm f(\gamma'(iy))$ for $\gamma' = [\begin{smallmatrix}
0 & -1 \\
N & 0 \\
\end{smallmatrix}]$ is enough.

If $f$ is an odd Maass form with expansion $\sum_{-\infty}^{\infty} a_n \sqrt{y} K_{\nu}(2\pi|n|y) e^{2\pi inx}$ with $a_0 = 0$ then $f(x+iy) = \sum_{1}^{\infty} a_n \sqrt{y} K_{\nu}(2\pi |n|y)i \sin(2\pi nx)$. The terms of $e^{2 \pi i n z}$ involving $\cos(2\pi nx)$ cancel for $a_n$ and $a_{-n}$ and hence are omitted. Then $f(iy) =  \sum_{1}^{\infty} a_n i \sqrt{y} K_{\nu}(2\pi |n|y) \text{sin}(0) = 0$ for all $y>0$.

If $f$ is an even form then $\frac{\partial f}{\partial x}$ is odd and vice versa. Hence, showing that $f(iy) =f(\gamma'(iy))$ for an even form and $\frac{\partial f}{\partial x}(iy) = \frac{\partial f}{\partial x}(\gamma'(iy))$ for an odd form is sufficient.

%Even case!
$\textbf{Even Case:}$ We know from Equation 1.9.10 from \cite{2} that if $f(x+iy)$ is an even form with Fourier expansion  $\sum_{-\infty}^{\infty}a_n \sqrt{y} K_{\nu}(2\pi |n|y) e^{2\pi i nx}$ with $a_0 = 0$, then 
\begin{equation}\int_{0}^{\infty} f(iy)y^{s-\frac{1}{2}} \frac{dy}{y} = \frac{1}{2} \pi^{-s}\Gamma\left(\frac{s + \nu}{2}\right) \Gamma\left(\frac{s - \nu}{2}\right) L(s,f)
\end{equation}
where $L(s,f)= \sum \frac{a_n}{n^s}$. Defining \begin{equation}
\Lambda (s,f) \coloneqq N^{(\frac{s-1/2}{2})} \pi^{-s} \Gamma\left(\frac{s+ \nu}{2}\right) \Gamma\left(\frac{s - \nu}{2}\right) L(s,f)
\end{equation}
we get $$\int_{0}^{\infty} f(iy)y^{s-\frac{1}{2}} \frac{dy}{y} = \frac{1}{2} N^{-(\frac{s-1/2}{2})} \Lambda(s,f).$$
Therefore,
$$f(iy) = 2\frac{1}{2\pi i}\int_{(c)} N^{-(\frac{s-1/2}{2})} \Lambda(s,f) y^{-(s-1/2)}ds$$ by the Mellin inversion formula 1.5.5 from \cite{2}.
If $\Lambda(s,f) = \Lambda(1-s,f)$, then
$$
f(iy) = 2\frac{1}{2\pi i}\int_{(c)} N^{-(\frac{s-1/2}{2})} \Lambda(1-s,f) y^{-(s-1/2)}ds.
$$
Let $s' = 1-s$, then $ds' = -ds$, $s-1/2$ = $1/2 - s'$ and $N^{-(\frac{s-1/2}{2})} = N^{-(\frac{s'-1/2}{2})} N^{(s'-1/2)}$.
\begin{align*}
\therefore f(iy) =& 2\frac{-1}{2\pi i}\int_{(c)} N^{-(\frac{s'-1/2}{2})} \Lambda(s',f) N^{(s'-1/2)}y^{(s'-1/2)}ds'
\\=& 2\frac{-1}{2\pi i}\int_{(c)} N^{-(\frac{s'-1/2}{2})} \Lambda(s',f) (Ny)^{(s'-1/2)}ds'
\\=& -2\frac{1}{2\pi i}\int_{(c)} N^{-(\frac{s-1/2}{2})} \Lambda(s,f) \left(\frac{1}{Ny}\right) ^{-(s-1/2)} ds
\\=& -f\left(\frac{i}{Ny}\right)
\\=& -f\left(\frac{-1}{iNy}\right)
\end{align*}
Therefore, $f(iy) = \pm f(\gamma'(iy))$ where $\gamma' = \begin{bmatrix}
0 & -1 \\
N & 0 \\
\end{bmatrix}$.

%Odd case!    
$\textbf{Odd Case:}$ If $f(z)$ is an odd form, then consider $g(z) = \frac{1}{4 \pi i}\frac{\partial f}{\partial x}(z)$.  In this case, we prove the relation $g(iy) = \pm \frac{1}{Ny^2}g(\gamma'(iy))$ which is essentially the equality of first partials. The Fourier expansion is $g(z) = \sum_{-\infty}^{\infty} n a_n  \sqrt{y} K_{\nu}(2\pi |n|y) e^{2\pi i nx}$. Therefore,
\begin{align*}
\int_0^{\infty} g(iy) y^{(s+1)-\frac{1}{2}}&\frac{dy}{y}\\ 
=&  \frac{1}{2}\pi^{-(s+1)} \Gamma\left(\frac{s+1+\nu}{2}\right) \Gamma\left(\frac{s+1-\nu}{2}\right)\sum_1^\infty \frac{na_n}{n^{s+1}}\\
=&\frac{1}{2}\pi^{-(s+1)} \Gamma\left(\frac{s+1+\nu}{2}\right) \Gamma\left(\frac{s+1-\nu}{2}\right)L(s,f)
\end{align*} 
where $L(s,f) = \sum_1^\infty 
\frac{a_n}{n^s}$ as before.

So define $$\Lambda (s,f) = N^{(\frac{s+1-1/2}{2})} \pi^{-(s+1)} \Gamma\left(\frac{s+1+ \nu}{2}\right) \Gamma\left(\frac{s+1 - \nu}{2}\right) L(s,f).$$ 
Then $$\int_{0}^{\infty} f(iy)y^{s+1-\frac{1}{2}} \frac{dy}{y} = \frac{1}{2} N^{-(\frac{s+1-1/2}{2})} \Lambda(s,f).$$
Therefore, 
$$g(iy) = 2\frac{1}{2\pi i}\int_{(c)} N^{-(\frac{s+1-1/2}{2})} \Lambda(s,f) y^{-(s+1-1/2)}ds$$ by the Mellin inversion formula from 1.5.5 from \cite{2}.
Now if $\Lambda(s,f) = - \Lambda(1-s,f)$, then
$$g(iy) = 2\frac{-1}{2\pi i}\int_{(c)} N^{-(\frac{s+1/2}{2})} \Lambda(1-s,f) y^{-(s+1/2)}ds.$$

Let $s' = 1-s$, then $ds' = -ds$, $s+1/2$ = $3/2 - s'$. Also, let $y' = \frac{1}{Ny}$. Therefore, $y = 1/Ny'$ and $Ny' = 1/y$
\begin{align*}
\therefore g(iy) =& 2\frac{-1}{2\pi i}\int_{(c)} N^{-(\frac{3/2 -s'}{2})} \Lambda(s',f) y^{(s'-3/2)}ds'
\\=& 2\frac{1}{2\pi i}\int_{(c)} N^{-(\frac{3/2 -s'}{2})} \Lambda(s',f) (Ny')^{(s'-3/2)}ds'
\\=& 2\frac{1}{2\pi i}\int_{(c)} N^{-(\frac{3/2-s'}{2})} \Lambda(s',f) (Ny')^{-(s+1/2)}(Ny')^2 ds'
\\=& 2\frac{1}{2\pi i}\int_{(c)} N^{-(\frac{3/2-s'}{2})} \Lambda(s',f) N^{-(s+1/2)}y'^{-(s+1/2)}(Ny')^2 ds'
\\=& 2\frac{1}{2\pi i}\int_{(c)} N^{-1}N^{-(\frac{s'+1/2}{2})} \Lambda(s',f) y'^{-(s+1/2)}(Ny')^2 ds
\\=& 2\frac{1}{2\pi i}\int_{(c)} N^{-(\frac{s'+1/2}{2})} \Lambda(s',f) \left(\frac{1}{Ny}\right)^{-(s+1/2)}N^{-1}\left(\frac{1}{y}\right)^2 ds
\\=& 2\frac{1}{2\pi i}\left(\frac{1}{Ny^2}\right) \int_{(c)} N^{-(\frac{s'+1/2}{2})} \Lambda(s',f) \left(\frac{1}{Ny}\right)^{-(s+1/2)} ds
\\=& \left(\frac{1}{Ny^2}\right) g\left(\frac{i}{Ny}\right)
\\=& \frac{1}{Ny^2} g\left(\frac{-1}{iNy}\right)
\end{align*} as required.
\end{proof}

\section{Discussion of possible application}

Let $$\zeta(s,P)  := \pi^{-2s}\Gamma\left( s+ \frac{\sqrt{-1}r}{2}\right) \Gamma\left( s- \frac{\sqrt{-1}r}{2}\right) \sum_{\beta \in S \setminus \{ 0 \}} A(\beta) \frac{P(\beta)}{|\beta|^{2s}}$$  where $P$ is a harmonic polynomial of degree $l$. Then $\zeta(s,P)$ converges for $Re(s) > \frac{l+4+k}{2}$. Let $\{ P_{l,\nu} \}_\nu$ be the basis of Harmonic polynomials of degree $l$ on $\mathbb{H}$. Then, Maass converse theorem implies $F$ with coefficients $A(B)$ belongs to $\mathcal{M}(\Gamma_T;r)$ if for all $l \in \Z$ and for all $ \nu$, the following 3 conditions are satisfied: 
\begin{enumerate}
	\item $\zeta(s,P_{l,\nu})$ has analytic continuation to whole complex plane,
	\item $\zeta(s,P_{l,\nu})$ is bounded on any vertical strip of the complex plane,
	\item the functional equation 
	$\zeta(2+l-s,P_{l,\nu}) = (-1)^l\zeta(s,\hat{P}_{l,\nu})$ 
	holds, where $\hat{P}_{l,\nu}(x) := P(\overline{x})$ for all $x \in \mathbb{H}$.
\end{enumerate}
We would like to use this fact with a suitable $P$ to garner information about $C(-N)$. However, for each $N$ there are a lot of $\beta$ with $|\beta|^2 = 2N$, so we need to make a smart choice for $P$. 

\textbf{First observation:} $l$ cannot be odd. For odd $l$, $P(\beta)=-P(-\beta)$ where as $A(\beta)=A(-\beta)$. Hence, $\zeta(s,P)=0$ for every odd degree harmonic $P$. 

\textbf{Second observation:} By Lemma $\ref{lem1}$,  $\beta$ can be primitive only when $d=1$ and $u=0$ if and only if $|\beta|^2 \equiv 2$ (mod 4). Hence, $u$ for $\beta$ is basically all the extra powers of 2 in $\beta$. Now $d$ for $\beta$ and $\overline{\beta}$ are equal. At the same time, $|\beta| = |\overline{\beta}|$, hence both $\beta$ and $\overline{\beta}$ have the same $u$. Therefore, they have the same absolute value, common odd divisor as well as power of $\varpi_2$. In other words, $A(\beta) = A(\overline{\beta})$.

With both of these observations, we can look for polynomials which might be suitable. However, even after testing various polynomials none of them appear to be helpful. Either too little terms are eliminated or no good terms remain and computation complexity increases too fast with every increase in degree. A heuristic reason for the lack of suitable polynomials is that the information we are trying to gain is too specific in the whole space. So this approach seems not to be working.



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\bibliographystyle{plain}
%\bibliography{Xbib}
\begin{thebibliography}{n}
\bibitem{And1} A. Andrianov, - \emph{Modular descent and the Saito-Kurokawa conjecture}, Invent. Math., 53 (3): 267–280, doi:10.1007/BF01389767	

\bibitem{Andr} A. Andrianov and V. Zhuravlev, - \emph{Modular forms and Hecke operators}, Providence, R.I.: American Mathematical Society, 1995. Print. Translations of Mathematical Monographs ; v. 145.

\bibitem{14} M. Asgari and  R. Schmidt, - \emph{Siegel modular forms and representations}, Manuscripta Math. 104 (2001), 173-200

\bibitem{5} A. I. Badulescu - \emph{Global Jaquet-Langlands correspondence, multiplicity one and classification of automorphic representations}, Invent. Math. 172 (2008), no. 2, 383–438. 

\bibitem{4} A. I. Badulescu and D. Renard - \emph{Unitary dual of GL(n) at archimedean places and global Jacquet-Langlands correspondence}, Compositio Math. 146 (2010), 1115-1164

\bibitem{2} D. Bump - \emph{Automorphic Forms and Representations}, Cambridge Studies in Advanced Mathematics, 55. Cambridge University Press, Cambridge, 1997. xiv+574 pp. ISBN: 0-521-55098-X 

\bibitem{11} C. Bushnell and G. Henniart - \emph{The local Langlands conjecture for GL(2)}, Grundlehren der Mathematischen Wissenschaften, vol. 335, Springer-Verlag, Berlin, 2006.

\bibitem{HC} Harish-Chandra - \emph{Automorphic forms on semisimple lie groups}, Berlin, New York: Springer-Verlag, 1968. Print. Lecture Notes in Mathematics (Springer-Verlag) ; 62.

\bibitem{16} T. Ikeda - \emph{On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n}, Annals of Mathematics, Second Series, Vol. 154, No. 3 (Nov., 2001), pp. 641-681 

\bibitem{17} W. Kohnen - \emph{Lifting modular forms of half-integral weight to Siegel modular forms of even genus}, Math. Ann. 322 (2002), no. 4, 787–809. 

\bibitem{18} W. Kohnen and H. Kojima - \emph{A Maass space in higher genus}, Compos. Math. 141 (2005), no. 2, 313–322.

\bibitem{Kuro} N. Kurokawa - \emph{Examples of eigenvalues of Hecke operators on Siegel cusp forms of degree two}, Invent. Math., 49 (2): 149–165, doi:10.1007/bf01403084

\bibitem{Maass} H. Maass - \emph{\"{U}ber eine Spezialschar von Modulformen zweiten Grades}, Invent. Math., 52 (1): 95–104, doi:10.1007/bf01389857

\bibitem{M2} H. Maass - \emph{Automorphe Funktionen von meheren Ver anderlichen und Dirchletsche Reihen}, Abh. Math. Semin. Univ. Hambg. 16(3-4) (1949), 72-100.

\bibitem{MNP} M. Muto, H. Narita and A. Pitale - \emph{Lifting to GL(2) over a division quaternion algebra and an explicit construction of CAP representations}, Nagoya Mathematical Journal, \textbf{222}, pp 137-185 (2016)

\bibitem{3} A. Pitale - \emph{Lifting from SL(2) to GSpin(1, 4)}, Int. Math. Res. Not. 63, 3919-3966 (2005)

\bibitem{PSS} A. Pitale, A. Saha and R. Schmidt -\emph{Local and global Maass relations}, Math. Z. (2017), 287(1-2):655-677. DOI:10.1007/s00209-016-1840-5

\bibitem{13} D. Prasad and A. Raghuram - \emph{Representation theory of \GL(n) over non-Archimedean local fields}, School on Automorphic Forms on GL(n), 159–205, ICTP Lect. Notes, 21, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2008.

\bibitem{7} P. Sarnak - \emph{Notes on Generalized Ramanujan Conjectures},  Clay Mathematics Proceedings, Volume 4, 2005

\bibitem{Sat} I. Satake - \emph{Theory of spherical functions on reductive groups over p-adic fields}, Inst. Hautes Etudes Sci. Publ. Math., 18 (1963) 5–69.

\bibitem{12} R. Schmidt - \emph{Some remarks on local newforms for GL(2)}, Documenta Math. 21 (2016), 467-553

\bibitem{Zag} D. Zagier, - \emph{Sur la conjecture de Saito-Kurokawa (d'après H. Maass)}, Seminar on Number Theory, Paris 1979–80, Progr. Math., 12, Boston, Mass.: Birkhäuser, pp. 371–394, MR 0633910
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