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 \(j\)-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 \(j \in \mathbb{N}\), we obtain asymptotics for the sums given by

$$\sum_{\substack{\alpha(\underline{x}))+1\le X \\ \underline{x} \in {\mathbb Z}^{4}}} \lambda_{ sym^{j}f}^{2}(\alpha(\underline{x})+1) ,\quad \sum_{\substack{\beta(\underline{x}))+1\le X \\ \underline{x} \in {\mathbb Z}^{4}}}\lambda_{ sym^{j}f}^{2}(\beta(\underline{x})+1)$$

, where \(\lambda_{ sym^{j}f}^{2}(n)\) denotes the coefficient of \(j\)-th symmetric power lift of classical Hecke eigenforms \(f\), the polynomials \(\alpha\) and \(\beta\) are given by

$$\alpha(\underline{x}) = \frac{1}{2} \big( x_{1}^{2}+ x_{1} + x_{2}^{2} + x_{2} + 2 ( x_{3}^{2} + x_{3}) + 4 (x_{4}^{2} + x_{4}) \big) \in \mathbb {Q}[x_{1},x_{2},x_{3},x_{4}], $$

and

$$\beta(\underline{x}) = x_{1}^{2} + \frac{x_{2}(x_{2} + 1)}{2} + \frac{x_{3}(x_{3}+1)}{2} + 6\cdot \frac{x_{4}( x_{4}+1)}{2} \in {\mathbb Q}[x_{1},x_{2},x_{3},x_{4}]$$