On the generation of the coefficient field of a newform by a single Hecke eigenvalue
Abstract
Let f be a nonCM 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 pth 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 nontrivial 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2007
 arXiv:
 arXiv:0711.3405
 Bibcode:
 2007arXiv0711.3405T
 Keywords:

 Mathematics  Number Theory;
 11F30 (Primary);
 11F11;
 11F25;
 11F80;
 11R45 (Secondary)
 EPrint:
 13 pages, more complete result, some corollaries added