On the generation of the coefficient field of a newform by a single Hecke eigenvalue
Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 2, p. 373-384

Let f be a non-CM newform of weight k2. 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 for further investigation.

Soit f une forme nouvelle de poids k2 sans multiplication complexe. Soit L un sous-corps du corps des coefficients de f. Nous résolvons complètement la question de la densité de l’ensemble des premier p tels que le p-ième coefficient de f engendre L. Cette densité est déterminée par les tordues intérieures de f. Comme cas particulier, on obtient que cette densité est 1 pour L le corps des coefficients de f, pourvu que f n’ait pas de tordue intérieure non-triviale. Nous présentons aussi quelques données nouvelles sur la réductibilité de polynômes de Hecke suggérant des questions pour des recherches à venir.

@article{JTNB_2008__20_2_373_0,
     author = {Koo, Koopa Tak-Lun and Stein, William and Wiese, Gabor},
     title = {On the generation of the coefficient field of a newform by a single Hecke eigenvalue},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {20},
     number = {2},
     year = {2008},
     pages = {373-384},
     doi = {10.5802/jtnb.633},
     mrnumber = {2477510},
     zbl = {1171.11027},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2008__20_2_373_0}
}
Koo, Koopa Tak-Lun; Stein, William; Wiese, Gabor. On the generation of the coefficient field of a newform by a single Hecke eigenvalue. Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 2, pp. 373-384. doi : 10.5802/jtnb.633. http://www.numdam.org/item/JTNB_2008__20_2_373_0/

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