Sums of k-th powers and Fourier coefficients of cusp forms

In this paper, we will first establish a power saving result for the shifted convolution sums of k-th powers and the normalized Fourier coefficients of SL2(Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {SL}}_2(\mathbb {Z} )$$\end{document} cusp forms. Later we will generalize the result to higher rank cases.