Higher moments of the Fourier coefficients of symmetric square L-functions on certain sequence

Abstract

Let $\lambda _{sym^{2}f}(n)$ be the $n^{th}$ normalized Fourier coefficient of symmetric square L-function. In this paper, we will establish an asymptotic formula for

$$\begin{aligned} \sum _{{\mathop {(a,b,c,d)\in \mathbb {Z}^{4}}\limits ^{a^{2}+b^{2}+c^{2}+d^{2}\le {x}}}}\lambda ^{\theta }_{sym^{2}f}(a^{2}+b^{2}+c^{2}+d^{2}) \end{aligned}$$

where $\theta =3,4$ and $x\ge {x_{0}}$ (sufficiently large).