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+cX14) 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 if(n)≪n14 for all n≫0