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$$sum_{nleq x}lambda_f(n^2) ll x^{frac{1}{2}}k^{frac{1}{2}}(log(x+k))^6$$is established, which improves the previous result.