On the gaps between non-zero Fourier coefficients of cusp forms of higher weight

Abstract

We show that if a modular cuspidal eigenform $f$ of weight $2k$ is $2$-adically close to an elliptic curve $E/\mathbb{Q}$, which has a cyclic rational $4$-isogeny, then $n$-th Fourier coefficient of $f$ is non-zero in the short interval $(X,X+c{X}^{\frac{1}{4}})$ for all $X\gg 0$ and for some $c>0$. We use this fact to produce non-CM cuspidal eigenforms $f$ of level $N>1$ and weight $k>2$ such that $i_f(n)\ll n^{\frac{1}{4}}$ for all $n\gg 0$.

arXiv commentTo appear in The Ramanujan Journal

References in corpus (1)

• Small gaps in coefficients of L-functions and B-free numbers in small intervals

Cited by in corpus (1)

• On the distribution of coefficients of half-integral weight modular forms and the Bruinier-Kohnen Conjecture