A short note on inadmissible coefficients of weight 2 and $$2k+1$$ newforms

Let $$f(z)=q+\sum _{n\ge 2}a(n)q^n$$ f ( z ) = q + ∑ n ≥ 2 a ( n ) q n 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 $$k=2$$ k = 2 by ruling out or locating all odd prime values $$|\ell |<100$$ | ℓ | < 100 of their Fourier coefficients a(n) when n satisfies some congruences. We also study the case of odd weights $$k\ge 1$$ k ≥ 1 newforms where the nebentypus is given by a quadratic Dirichlet character.