Bounds for twisted symmetric square $L$-functions

Let $f\in S_k(N,ψ)$ be a newform, and let $χ$ 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 $$ L(\tfrac{1}{2},\Sym f\otimesχ)\ll_{f,\varepsilon} q^{3/4\ell-δ_{\ell}+\varepsilon} $$ for some $δ_{\ell}>0$ and any $\varepsilon>0$. In particular, for $\ell=3$ we show that the bound holds with $δ_{\ell}=1/4$.

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