Two properties of symmetric cube transfers of modular forms

In this article, we study two important properties of ${\mathrm{s}\mathrm{y}\mathrm{m}}^3$ transfers of the automorphic representation $\pi$ associated to a modular form. First we compute the conductor of ${\mathrm{s}\mathrm{y}\mathrm{m}}^3(\pi )$. Then we detect the types of local automorphic representations at bad primes by the variation of the epsilon factors of symmetric cube transfer of the representation $\pi$ attached to a cusp form $f$. Here we twist the modular forms by a specific quadratic character. From this variation number, for each prime $p$, we classify all possible types of symmetric cube transfers of the local representations ${\pi }_p$. For ${\mathrm{s}\mathrm{y}\mathrm{m}}^3$ transfer, the most difficult prime is $p=3$.