ON THE NUMBER OF SIGN CHANGES OF HECKE EIGENVALUES OF NEWFORMS

Abstract

AbstractWe show that, for every x exceeding some explicit bound depending only on k and N, there are at least C(k,N)x/log 17x 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.