On the gaps between nonzero Fourier coefficients of eigenforms with CM

Abstract

Suppose [Formula: see text] is an elliptic curve over [Formula: see text] of conductor [Formula: see text] with complex multiplication (CM) by [Formula: see text], and [Formula: see text] is the corresponding cuspidal Hecke eigenform in [Formula: see text]. Then [Formula: see text]th Fourier coefficient of [Formula: see text] is nonzero in the short interval [Formula: see text] for all [Formula: see text] and for some [Formula: see text]. As a consequence, we produce infinitely many cuspidal CM eigenforms [Formula: see text] level [Formula: see text] and weight [Formula: see text] for which [Formula: see text] holds, for all [Formula: see text].