Average behaviour of Fourier coefficients of j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j$$\end{document}-symmetric power L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L$$\end{document}-functions over some polynomials

We establish the asymptotics of the second moment of the coefficient of j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j$$\end{document}-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∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in \mathbb{N}$$\end{document}, we obtain asymptotics for the sums given by ∑α(x̲))+1≤Xx̲∈Z4λsymjf2(α(x̲)+1),∑β(x̲))+1≤Xx̲∈Z4λsymjf2(β(x̲)+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum_{\substack{\alpha(\underline{x}))+1\le X \\ \underline{x} \in {\mathbb Z}^{4}}} \lambda_{ sym^{j}f}^{2}(\alpha(\underline{x})+1) ,\quad \sum_{\substack{\beta(\underline{x}))+1\le X \\ \underline{x} \in {\mathbb Z}^{4}}}\lambda_{ sym^{j}f}^{2}(\beta(\underline{x})+1)$$\end{document}, where λsymjf2(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda_{ sym^{j}f}^{2}(n)$$\end{document} denotes the coefficient of j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j$$\end{document}-th symmetric power lift of classical Hecke eigenforms f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f$$\end{document}, the polynomials α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document} and β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document} are given by α(x̲)=12(x12+x1+x22+x2+2(x32+x3)+4(x42+x4))∈Q[x1,x2,x3,x4],\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha(\underline{x}) = \frac{1}{2} \big( x_{1}^{2}+ x_{1} + x_{2}^{2} + x_{2} + 2 ( x_{3}^{2} + x_{3}) + 4 (x_{4}^{2} + x_{4}) \big) \in \mathbb {Q}[x_{1},x_{2},x_{3},x_{4}], $$\end{document} and β(x̲)=x12+x2(x2+1)2+x3(x3+1)2+6·x4(x4+1)2∈Q[x1,x2,x3,x4]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta(\underline{x}) = x_{1}^{2} + \frac{x_{2}(x_{2} + 1)}{2} + \frac{x_{3}(x_{3}+1)}{2} + 6\cdot \frac{x_{4}( x_{4}+1)}{2} \in {\mathbb Q}[x_{1},x_{2},x_{3},x_{4}]$$\end{document}