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

2007/11/21 by Koo, Koopa Tak-Lun, Stein, William, Wiese, Gabor · 1 citation
#11F11 #11F25 #11F30 (Primary) #11F80 #11R45 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0711.3405

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.

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• Classifications of Hecke Fields and Galois Images of Weight One Exotic Newforms

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