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In this article, we study two important properties of ${\rm{sym}}^3$transfers of the automorphic representation $\pi$ associated to a modular form.First we compute the conductor of ${\rm{sym}}^3(\pi)$. Then we detect the typesof local automorphic representations at bad primes by the variation of theepsilon factors of symmetric cube transfer of the representation $\pi$ attachedto a cusp form $f$. Here we twist the modular forms by a specific quadraticcharacter. From this variation number, for each prime $p$, we classify allpossible types of symmetric cube transfers of the local representations$\pi_p$. For ${\rm{sym}}^3$ transfer, the most difficult prime is $p=3$.

Two properties of symmetric cube transfers of modular forms