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

Abstract

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