Estimates for Fourier Coefficients of Cusp Forms in Weight Aspect

Let $f$ be a holomorphic Hecke eigenform of weight $k$ for the modular group $\Gamma=SL_2(\mathbb{Z})$ and let $\lambda_f(n)$ 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 $$ \sum_{n\leq x}\lambda_f(n^2) \ll x^{\frac{1}{2}}k^{\frac{1}{2}}(\log (x+k))^6 $$ is established, which improves the previous result.