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Dihedral Galois representations and Katz modular forms.

Wiese, Gabor — 2004

Documenta Mathematica

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

Koopa Tak-Lun Koo; William Stein; Gabor Wiese — 2008

Journal de Théorie des Nombres de Bordeaux

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