A lower bound for the discrepancy in a Sato-Tate type measure

Abstract

Let Sk(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_k(N)$$\end{document} denote the space of cusp forms of even integer weight k and level N. We prove an asymptotic for the Petersson trace formula for Sk(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_k(N)$$\end{document} under an appropriate condition. Using the non-vanishing of a Kloosterman sum involved in the asymptotic, we give a lower bound for discrepancy in the Sato-Tate distribution for levels not divisible by 8. This generalizes a result of Jung and Sardari (Math Ann 378(1-2):513-557, 2020, Theorem 1.6) for squarefree levels. An analogue of the Sato-Tate distribution was obtained by Omar and Mazhouda (Ramanujan J 20(1):81-89, 2009, Theorem 3) for the distribution of eigenvalues lambda p2(f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{p<^>2}(f)$$\end{document} where f is a Hecke eigenform and p is a prime number. As an application of the above-mentioned asymptotic, we obtain a sequence of weights kn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_n$$\end{document} such that discrepancy in the analogue distribution obtained in Omar and Mazhouda (Ramanujan J 20(1):81-89, 2009) has a lower bound