Applications of the petersson formula for a bilinear form in fourier coefficients of cusp forms

Abstract

Let Sk(Γ0(N),χ) be the space of holomorphic Γ0(N) forms of integral weight k and character χ. Let fj(z), 1≤j≤v new2k (p), be the set of normalized newforms of S2k(Γ0(p),1), where p is a prime, and let\(L_j (s) = L_{f_i } (s)\) be the L-function of fj(z). It is proved that $$\sum\limits_{1 \leqslant j \leqslant v_{2k}^{new} (p)} {L_j^2 (\tfrac{1}{2}) \ll p\log ^4 p \cdot \log \log p} , p \to \infty$$ where 2k≥4. Errors in an earlier paper (RŽMat, 1989, 4A65) are corrected. Bibliography: 11 titles.