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

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 $requiv 0 (textrm{mod} 3)$. Then we have [ Lleft( f otimes chi, frac{1}{2}right) ll_{f, epsilon} P^{1/3 +epsilon}, ] 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 [ Lleft( f otimes chi, frac{1}{2}right) ll_{f, epsilon} P^{3/8 +epsilon}, ] where $epsilon > 0$ is any positive real number.