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

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 $S_2^{\mathrm{n}\mathrm{e}\mathrm{w}}({\mathrm{\Gamma }}_0(N))$. Then $n$-th Fourier coefficient of $f_E$ is non-zero in the short interval $(X, X + cX^{\frac{1}{4}})$forall$X \gg 0$andforsome$c > 0$. As a consequence, we produce infinitely many cuspidal CM eigenforms $f$ level $N>1$ and weight $k > 2$forwhich$i_f(n) \ll n^{\frac{1}{4}}$holds,forall$n \gg 0$.