Weighted average values of automorphic $L$-functions

Abstract

Let $S_2^*(q)$ be the set of primitive Hecke eigenforms of weight 2 and prime level $q$. For $p$ prime and $t\in \mathbb{R}$, we prove asymptotic formulas for the sums $$ \mathcal {A}(p^j,q,t)=\sum_{f\in S_2^*(q)} L\left(\frac{1}{2}+it,f\right)^2\lambda_f(p^j),\qquad j=1,2, $$ where $\lambda_f(p^j)$ is the $p^j$-th normalized Fourier coefficient of $f$ and $L(s,f)$ is the $L$-function associated to $f$.