On the generation of the coefficient field of a newform by a single Hecke eigenvalue

TL;DR

This paper investigates when a single Hecke eigenvalue of a non-CM newform generates its coefficient field, linking the density of such primes to the form's inner twists, and provides new data on Hecke polynomial reducibility.

Contribution

It completely characterizes the density of primes where a newform's coefficient generates a subfield, based on the form's inner twists, and offers new insights into Hecke polynomial reducibility.

Findings

Density of primes generating the coefficient field depends on inner twists.

In absence of non-trivial inner twists, the density is 1.

New data on reducibility of Hecke polynomials suggest further questions.

Abstract

Let f be a non-CM newform of weight k > 1. Let L be a subfield of the coefficient field of f. We completely settle the question of the density of the set of primes p such that the p-th coefficient of f generates the field L. This density is determined by the inner twists of f. As a particular case, we obtain that in the absence of non-trivial inner twists, the density is 1 for L equal to the whole coefficient field. We also present some new data on reducibility of Hecke polynomials, which suggest questions for further investigation.