Sign changes of Fourier coefficients of entire modular integral

Abstract

Let f be a non-zero cusp form with real Fourier coefficients a(n) (n >= 1) of positive real weight k and a unitary multiplier system upsilon on a subgroup Gamma subset of SL2(R) that is finitely generated and of Fuchsian type of the first kind. Then, it is known that the sequence (a(n))(n >= 1) has infinitely many sign changes. In this short note, we generalise the above result to the case of entire modular integrals of non-positive integral weight k on the group Gamma*(0)(N) (N is an element of N) generated by the Hecke congruence subgroup Gamma(0)(N) and the Fricke involution W-N := ((0)(root N) (-1/root N)(0)) provided that the associated period functions are polynomials.