Let f(z)=q+∑n≥2a(n)qn be a weight k normalized newform with integer coefficients and trivial residual mod 2 Galois representation. We extend the results of Amir and Hong in Amir and Hong (On L-functions of modular elliptic curves and certain K3 surfaces, Ramanujan J, 2021) for k=2 by ruling out or locating all odd prime values |ℓ|<100 of their Fourier coefficients a(n) when n satisfies some congruences. We also study the case of odd weights k≥1 newforms where the nebentypus is given by a quadratic Dirichlet character.