Room 326

on the gaps between non-zero Fourier coefficinets of cusp forms of higher weight and level

Narasimha Kumar

IIT Hyderabad

We show that if a modular cuspidal eigenform $f$ of weight $2k$ is $2$-adically close to an elliptic curve $E/\Q$,
which has a cyclic rational $4$-isogeny, then $n$-th Fourier coefficient of $f$ is non-zero in the short interval $(X, X + cX^{\frac{1}{4}})$
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 $n \gg 0$.