Weighted average values of automorphic $L$-functions

Abstract

Let $S_2^{\ast }(q)$ 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 $$\mathcal{A}(p^j,q,t)=\sum _{f\in S_2^{\ast }(q)}L{\left(\frac{1}{2}+it,f\right)}^2{\lambda }_f(p^j),j=1,2,$$ 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 $f$.

arXiv comment19 pages