Higher power sums of Fourier coefficients of symmetric square <i>L</i>-functions

Abstract Let f ( z ) be the normalized primitive holomorphic Hecke eigenforms of even integral weight k for the full modular group $SL(2,\mathbb{Z})$ and denote $L(s,\mathrm{sym}^{2}f)$ be the symmetric square L -function attached to f ( z ). Suppose that $\lambda_{\mathrm{sym}^{2}f}(n)$ be the $\mathrm{Fourier}$ coefficient of $L(s,\mathrm{sym}^{2}f)$ . In this paper, we investigate the sum $\sum\limits_{n\leqslant x} \lambda^{j}_{\mathrm{sym}^{2}f }(n) $ for $j\geqslant 3$ and obtain some new results which improve on previous error estimates. We also consider the sum $\sum\limits_{n\leqslant x}\lambda^{j}_{f }(n^{2})$ and get some similar results.