Applications of the Petersson formula for a bilinear form in Fourier coefficients of cusp forms

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 11, Tome 204 (1993), pp. 143-166

Résumé

Let $S_{2k}(\Gamma_0(N),\chi)$ be the space of holomorphic $\Gamma_0(N)$-cusp forms of integral weight $k$ and character $\chi$. Let $f_j(z)$, $1\le j\le v_{2k}^\mathrm{new}(p)$, be the set of normalized newforms of $S_{2k}(\Gamma_0(p),1)$, where $p$ is a prime, and let $L_j(s)=L_{f_j}(s)$ be the $L$-function of $f_j(z)$. It is proved that $$ \sum_{1\le j\le v_{2k}^\mathrm{new}(p)}L_j^2\left(\frac12\right)\ll p\log^4p\cdot\log\log p,\qquad p\to\infty, $$ where $2k\ge4$. Errors in an earlier paper (RŽMat, 1989, 4A65) are corrected. Bibliography: 11 titles.