Hybrid Subconvexity Bound for $L\left(\frac{1}{2},{\mathrm{Sym}}^{2}f\otimes \rho \right)$ via the Delta Method

Abstract

Let $P$ be a prime and $k$ be an even integer. Let $f$ be a full level holomorphic cusp form of weight $k$ and $\rho$ be a primitive level $P$ holomorphic cusp form with arbitrary nebentypus and fixed weight $\kappa$. We prove a hybrid subconvexity bound for $L\left(\frac{1}{2},{\mathrm{Sym}}^{2}f\otimes \rho \right)$ when $P^{\frac{1}{4}+\eta }<k<P^{\frac{21}{17}-\eta }$ for any $0<\eta <\frac{67}{136}$. This extends the range of $P$ and $k$ achieved by Holowinsky, Munshi and Qi. The result is established using a new variant of the delta method.