Estimates for fourier coefficients of cusp forms in weight aspect

Abstract

Let f be a holomorphic Hecke eigenform of weight k for the modular group Γ = SL ₂(Z) and let λ _f (n) be the n-th normalized Fourier coefficient. In this paper, by a new estimate of the second integral moment of the symmetric square L-function related to f, the estimate $\sum\limits_{n \leqslant x} {{\lambda _f}\left( {{n^2}} \right)} \ll {x^{\frac{1}{2}}}{k^{\frac{1}{2}}}{\left( {\log \left( {x + k} \right)} \right)^6}$ is established, which improves the previous result.

Keywords

Fourier coefficients / Cusp forms / Symmetric square L-function

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