Bounds for twisted symmetric square $L$-functions

Abstract

Let $f\in S_k(N,\psi)$ be a newform, and let $\chi$ be a primitive character of conductor $q^{\ell}$. Assume that $q$ is a prime and $\ell>1$. In this paper we describe a method to establish convexity breaking bounds of the form $$ L(\tfrac{1}{2},\Sym f\otimes\chi)\ll_{f,\varepsilon} q^{3/4\ell-\delta_{\ell}+\varepsilon} $$ for some $\delta_{\ell}>0$ and any $\varepsilon>0$. In particular, for $\ell=3$ we show that the bound holds with $\delta_{\ell}=1/4$.