Paramodular forms coming from elliptic curves

Abstract

There is a lifting from a non-CM elliptic curve $E/\mathbb{Q}$ to a paramodular form $f$ of degree $2$ and weight $3$ given by the symmetric cube map. We find the level of $f$ in an explicit way in terms of the coefficients of the Weierstrass equation of $E$. In order to compute the paramodular level, we use the available description of the local representations of $\mathrm{GL}(2,\mathbb{Q}_p)$ attached to $E$ for $p \ge 5$ and determine the local representation of $\mathrm{GL}(2,\mathbb{Q}_3)$ attached to $E$.