On the Number of Sign Changes of Hecke Eigenvalues of Newforms

Abstract

We show that, for every x exceeding some explicit bound depending only on k and N , there are at least C(k, N )x/ log 17 x positive and negative coefficients a(n) with n ≤ x in the Fourier expansion of any non-zero cuspidal Hecke eigenform of even integral weight k ≥ 2 and squarefree level N that is a newform, where C(k, N ) depends only on k and N . From this we deduce the existence of a sign change in a short interval.2000 Mathematics subject classification: 11F11, 11F30.

Fourier coefficients of cusp forms of half-integral weight

“…Owing to different reasons, the problem of sign changes of Hecke eigenvalues of integral weight cusp forms has attracted many attentions [10,15,7,14,19,17]. One motivation is to delve the analogy with (real) Dirichlet characters.…”

Sign changes of Fourier coefficients of Maass eigenforms

“…The more precise question, that how long is the sequence of Hecke eigenvalues that keep the same sign, has been studied in Kohnen-Lau-Shparlinski [7], in Wu [17], and in Lau-Wu [10]; in the last paper, best known result is published.…”

SIGN CHANGES OF THE COEFFICIENTS OF AUTOMORPHIC L-FUNCTIONS

“…• the distribution of Hecke eigenvalues. In the past few years, together with our collaborators W. Kohnen [49,50,40,51,54,43,53,46,47,41,83]. This survey gives an account of background and the recent development, which includes the great contributions of other authors.…”