On the gaps between non-zero Fourier coefficients of eigenforms with CM

Abstract

Suppose $E$ is an elliptic curve over $\mathbb{Q}$ of conductor $N$ with complex multiplication (CM) by $\mathbb{Q}(i)$, and $f_E$ is the corresponding cuspidal Hecke eigenform in $S_2^{\mathrm{new}}({\Gamma }_0(N))$. Then $n$-th Fourier coefficient of $f_E$ is non-zero in the short interval $(X,X+c{X}^{\frac{1}{4}})$ for all $X\gg 0$ and for some $c>0$. As a consequence, we produce infinitely many cuspidal CM eigenforms $f$ level $N>1$ and weight $k>2$ for which $i_f(n)\ll n^{\frac{1}{4}}$ holds, for all $n\gg 0$.