W\lowercase{eyl} \lowercase {bound for $p$-power twist of} $GL(2)$ L-\lowercase{functions }

Abstract

Let $f$ be a cuspidal eigenform (holomorphic or Maass) on the full modular group $SL(2,\mathbb{Z})$ . Let $\chi$ be a primitive character of modulus $P$. We shall prove the following results: 1. Suppose $P=p^r$, where $p$ is a prime and $r\equiv 0(\text{mod}3)$. Then we have \[ L\left( f \otimes χ, \frac{1}{2}\right) \ll_{f, ε} P^{1/3 +ε}, \] where $\epsilon >0$ is any positive real number. 2. Suppose $\chi$ factorizes as $\chi ={\chi }_1{\chi }_2$, where ${\chi }_i$'s are primitive character modulo $P_i$, where $P_i$ are primes, $P^{1/2-\epsilon }\ll P_i\ll P^{1/2+\epsilon }$ for $i=1,2$ and $P=P_1{P}_2$. We have the Burgess bound \[ L\left( f \otimes χ, \frac{1}{2}\right) \ll_{f, ε} P^{3/8 +ε}, \] where $\epsilon >0$ is any positive real number.