Average behaviour of Fourier coefficients of \(j\)-symmetric power \(L\)-functions over some polynomials

We establish the asymptotics of the second moment of the coefficient of \(j\) $j$ -th symmetric poower lift of classical Hecke eigenforms over certain polynomials, given by a sum of triangular numbers with certain positive coefficients. More precisely, for each \(j \in \mathbb{N}\) $j\in \mathbb{N}$ , we obtain asymptotics for the sums given by \(\sum_{\substack{\alpha(\underline{x}))+1\le X \\ \underline{x} \in {\mathbb Z}^{4}}}\lambda_{ sym^{j}f}^{2}(\alpha(\underline{x})+1) ,\quad \sum_{\substack{\beta(\underline{x}))+1\le X \\ \underline{x} \in {\mathbb Z}^{4}}}\lambda_{ sym^{j}f}^{2}(\beta(\underline{x})+1)\) $$\sum _{\begin{array}{c}\alpha (\underline{x}))+1\le X\\ \underline{x}\in {\mathbb{Z}}^4\end{array}}{\lambda }_{sy{m}^{j}f}^2(\alpha (\underline{x})+1),\sum _{\begin{array}{c}\beta (\underline{x}))+1\le X\\ \underline{x}\in {\mathbb{Z}}^4\end{array}}{\lambda }_{sy{m}^{j}f}^2(\beta (\underline{x})+1)$$ ,where \(\lambda_{ sym^{j}f}^{2}(n)\) ${\lambda }_{sy{m}^{j}f}^2(n)$ denotes the coefficient of \(j\) $j$ -th symmetric power lift of classical Hecke eigenforms \(f\) $f$ , the polynomials \(\alpha\) $\alpha$ and \(\beta\) $\beta$ are given by \(\alpha(\underline{x}) = \frac{1}{2} \big( x_{1}^{2}+ x_{1} + x_{2}^{2} + x_{2} + 2 ( x_{3}^{2} + x_{3}) + 4 (x_{4}^{2} + x_{4}) \big) \in \mathbb {Q}[x_{1},x_{2},x_{3},x_{4}],\) $$\alpha (\underline{x})=\frac{1}{2}(x_1^2+x_1+x_2^2+x_2+2(x_3^2+x_3)+4(x_4^2+x_4))\in \mathbb{Q}[x_1,x_2,x_3,x_4],$$ and \(\beta(\underline{x}) = x_{1}^{2} + \frac{x_{2}(x_{2} + 1)}{2} + \frac{x_{3}(x_{3}+1)}{2} + 6\cdot \frac{x_{4}( x_{4}+1)}{2} \in {\mathbb Q}[x_{1},x_{2},x_{3},x_{4}]\) $$\beta (\underline{x})=x_1^2+\frac{x_2(x_2+1)}{2}+\frac{x_3(x_3+1)}{2}+6·\frac{x_4(x_4+1)}{2}\in \mathbb{Q}[x_1,x_2,x_3,x_4]$$