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

TL;DR

This paper investigates the specific Fourier coefficients of weight 2 and odd weight newforms with integer coefficients, extending previous results to identify or exclude certain prime values based on congruence conditions.

Contribution

It generalizes earlier work by Amir and Hong for weight 2 newforms and explores the behavior of Fourier coefficients for odd weights with quadratic nebentypus.

Findings

Identified all odd prime coefficients less than 100 for certain newforms.

Extended the classification of Fourier coefficients to odd weights with quadratic nebentypus.

Ruled out or located specific prime values of Fourier coefficients under certain conditions.

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.