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

In this paper, we investigate the average behavior of the $n^{th}$ normalized Fourier 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 primitive holomorphic cusp form $f$ of weight $k$ for the full modular group $SL(2,\mathbb{Z})$ over a certain sequences of positive integers. Precisely, we prove 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