Bounds for twisted symmetric square L-functions

Abstract

Let f ∈ S_k(N,ψ) be a newform, and let χ be a primitive character of conductor q^ℓ. Assume that q is a prime and ℓ > 1. In this paper we describe a method to establish convexity breaking bounds of the form L(1/2, Sym²f⊗χ)_f,εq^{3/4ℓ-δ_ℓ+ε} for some δ_ℓ > 0 and any ε > 0. In particular, for we show that the bound holds with δ_ℓ = 1/4.