Simultaneous sign change of Fourier-coefficients of two cusp forms

Abstract

We consider the simultaneous sign change of Fourier coefficients of two modular forms with real Fourier coefficients. In an earlier work, the second author with Sengupta proved that two cusp forms of different (integral) weights with real algebraic Fourier coefficients have infinitely many Fourier coefficients of the same as well as opposite sign, up to the action of a Galois automorphism. In the first part, we strengthen their result by doing away with the dependency on the Galois conjugacy. In fact, we extend their result to cusp forms with arbitrary real Fourier coefficients. Next we consider simultaneous sign change at prime powers of Fourier coefficients of two integral weight Hecke eigenforms which are newforms. Finally, we consider an analogous question for Fourier coefficients of two half-integral weight Hecke eigenforms.