  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 $S^{\mathrm{new}}_2(\Gamma_0(N))$. Then $n$-th
Fourier coefficient of $f_E$ is non-zero in the short interval $(X, X +
cX^{\frac{1}{4}})$ 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 $n \gg 0$.

Number Theory