On the change of epsilon factors for symmetric square transfers under twisting and applications

Let us consider the symmetric square transfer of the automorphic representation $\pi$ associated to a modular form $f\in S_k(N,\epsilon )$. In this article, we study the variation of the epsilon factor of ${\mathrm{s}\mathrm{y}\mathrm{m}}^2(\pi )$ under twisting in terms of the local Weil-Deligne representation at each prime $p$. As an application, we detect the possible types of the symmetric square transfer of the local representation at $p$. Furthermore, as the conductor of ${\mathrm{s}\mathrm{y}\mathrm{m}}^2(\pi )$ is involved in the variation number, we compute it in terms of $N$.