There is a lifting from a non-CM elliptic curve E/Q to a cuspidal paramodular newform f of degree 2 and weight 3 given by the symmetric cube map. We find a description of the level of f in terms of the coefficients of the Weierstrass equation of E. In order to compute the paramodular level, we need a detailed description of the local representations πp of \GL(2,Qp) attached to E/Qp, where π≅⨂pπp is the cuspidal automorphic representation of \GL(2,AQ) associated with E/Q. We use the available description of the local representations of \GL(2,Qp) attached to E for p≥5 and determine the local representation of \GL(2,Q3) attached to E. In fact, we study the representations of \GL(2,K) attached to E/K for any non-archimedean local field K of characteristic 0 and residue characteristic 3.