The average behaviour of Hecke eigenvalues over certain sparse sequence of positive integers

Let j≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\ge 2$$\end{document} be a given integer. Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group Γ=SL(2,Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma =SL(2,\mathbb {Z})$$\end{document}. Denote by λsymjf(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{\text {sym}^{j}f}(n)$$\end{document} the nth normalized coefficient of the Dirichlet expansion of the jth symmetric power L-function L(symjf,s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(\text {sym}^{j}f,s)$$\end{document} attached to f. In this paper, we are interested in the average behaviour of the following sum ∑a2+b2+c2+d2≤x(a,b,c,d)∈Z4λsymjf2(a2+b2+c2+d2),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{\begin{array}{c} a^{2} + b^{2} + c^{2} + d^{2}\le x \\ (a,b,c,d)\in \mathbb {Z}^{4} \end{array}} \lambda _{\text {sym}^{j}f}^{2}(a^{2}+b^{2}+c^{2}+d^{2}), \end{aligned}$$\end{document}where x is sufficiently large, which improves and generalizes the recent work of Sharma and Sankaranarayanan [52]. By analogy, we also consider the analogous results for higher moments of normalized Fourier coefficients and the second moment of normalized coefficients of two symmetric power L-functions attached to two distinct cusp forms of the same sequence.