Fourier coefficients of cusp forms of half-integral weight

Abstract

Let \({{\mathfrak f}}\) be a cusp form of half integral weight whose Fourier coefficients \({{\mathfrak a}_{\mathfrak f}(n)}\) are all real. We study the sign change problem of \({{\mathfrak a}_{\mathfrak f}(n)}\) , when n runs over some specific sets of integers. Lower bounds of the best possible order of magnitude are established for the number of those coefficients that have the same signs. These give an improvement on some recent results of Bruinier and Kohnen (Modular forms on Schiermonnikoong. Cambridge University Press, Cambridge 57–66, 2008) and Kohnen (Int. J. Number. Theory 6:1255–1259, 2010).