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

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(1/4)) for all X >> 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) << n(1/4) for all n >> 0.