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In this paper, we investigate the average behavior of the $n^{th}$ normalizedFourier coefficients of the $j^{th}$ ($j \geq 2$ be any fixed integer)symmetric power $L$-function (i.e., $L(s,sym^{j}f)$), attached to a primitiveholomorphic cusp form $f$ of weight $k$ for the full modular group$SL(2,\mathbb{Z})$ over a certain sequences of positive integers. Precisely, weprove an asymptotic formula with an error term for the sum$$\sum_{\stackrel{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}\leq{x}}{(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})\in\mathbb{Z}^{6}}}\lambda^{2}_{sym^{j}f}(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}),$$where $x$ is sufficiently large, and$$L(s,sym^{j}f):=\sum_{n=1}^{\infty}\dfrac{\lambda_{sym^{j}f}(n)}{n^{s}}.$$When $j=2$, the error term which we obtain, improves the earlier known result.

On the average behavior of the Fourier coefficients of $j^{th}$ symmetric power $L$-function over a certain sequences of positive integers