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

Koo, Koopa Tak-Lun; Stein, William; Wiese, Gabor

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

Authors

Koo, Koopa Tak-Lun

Stein, William

Wiese, Gabor

Description

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.
13 pages, more complete result, some corollaries added