On the sign changes and non-vanishing of Hecke eigenvalues associated to symmetric power L-Functions

Let f be a Hecke cusp form of even integral weight k for the full modular group 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}$$SL(2,{\mathbb {Z}})$$\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 associated with f. In this paper, we derive a upper bound for the first negative coefficient for sequence {λsymjf(n)}n≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\lambda _{\text {sym}^{j}f}(n)\}_{n\ge 1}$$\end{document} in terms of the weight of f. Furthermore, we also investigate the length of the maximal string of terms with the same sign in an interval [1, x].