Congruences involving the Ul\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\ell }$$\end{document} operator for weakly holomorphic modular forms

Abstract

Let. be an integer, and f (z) = similar to n similar to-8 a(n)qn be a weakly holomorphic modular form of weight. + 1 2 on similar to 0(4) with integral coefficients. Let similar to = 5 be a prime. Assume that the constant term a(0) is not zero modulo similar to. Further, assume that, for some positive integer m, the Fourier expansion of ( f |U similar to m)(z) = similar to 8 n=0 b(n)qn has the form