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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.
Let $f $ be a holomorphic Hecke eigenform or a Hecke-Maass cusp form for the full modular group $ SL(2, mathbb{Z})$. In this paper we shall use circle method to prove the Weyl exponent for $GL(2)$ $L$-functions. We shall prove that [ L left( frac
Let $f $ be a holomorphic Hecke eigenforms or a Hecke-Maass cusp form for the full modular group $ SL(2, mathbb{Z})$. In this paper we shall use circle method to prove the Weyl exponent for $GL(2)$ $L$-functions. We shall prove that [ L left( fra
Let $phi$ be a Hecke-Maass cusp form for $SL(3, mathbb{Z})$ with Langlands parameters $({bf t}_{i})_{i=1}^{3}$ satisfying $$|{bf t}_{3} - {bf t}_{2}| leq T^{1-xi -epsilon}, quad , {bf t}_{i} approx T, quad , , i=1,2,3$$ with $1/2 < xi <1$ and any $ep
Let $F$ be a $G L(3)$ Hecke-Maass cuspform of level $P_1$ and $f$ be a $G L(2)$ Hecke-Maass cuspform of level $P_2$. In this article, we will prove a subconvex bound for the $G L(3) times G L(2)$ Rankin-Selberg $L$-function $L(s,Ftimes f)$ in the lev
Let $ mathfrak{f} $ run over the space $ H_{4k} $ of primitive cusp forms of level one and weight $ 4k $, $ k in N $. We prove an explicit formula for the mixed moment of the Hecke $ L $-function $ L(mathfrak{f}, 1/2) $ and the symmetric square $L$-f