Weyl bound for $p$-power twist of $\mathrm{GL}(2)$ $L$-functions

Abstract

Let $f$ be a cuspidal eigenform (holomorphic or Maass) for the congruence group ${\mathrm{\Gamma }}_0\left(N\right)$ with $N$ square-free. Let $p$ be a prime and let $\chi$ be a primitive character of modulus $p^{3r}$. We shall prove the Weyl-type subconvex bound

$$L\left(\frac{1}{2}+it,f\otimes \chi \right){\ll }_{f,t,\epsilon }{p}^{r+\epsilon },$$

where $\epsilon >0$ is any positive real number.