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. © 2016, Springer Science+Business Media New York.