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

Koopa Tak-Lun KooWilliam SteinGabor Wiese — 2008

Journal de Théorie des Nombres de Bordeaux

Let f be a non-CM newform of weight k 2 . 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 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...

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