Let be a non-CM newform of weight . Let be a subfield of the coefficient field of . We completely settle the question of the density of the set of primes such that the -th coefficient of generates the field . This density is determined by the inner twists of . As a particular case, we obtain that in the absence of nontrivial inner twists, the density is for equal to the whole coefficient field. We also present some new data on reducibility of Hecke polynomials, which suggest questions for further investigation.
Soit une forme nouvelle de poids sans multiplication complexe. Soit un sous-corps du corps des coefficients de . Nous résolvons complètement la question de la densité de l’ensemble des premier tels que le -ième coefficient de engendre . Cette densité est déterminée par les tordues intérieures de . Comme cas particulier, on obtient que cette densité est pour le corps des coefficients de , pourvu que 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 = {Koopa Tak-Lun Koo and William Stein and Gabor Wiese}, 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}, pages = {373--384}, publisher = {Universit\'e Bordeaux 1}, volume = {20}, number = {2}, year = {2008}, doi = {10.5802/jtnb.633}, mrnumber = {2477510}, zbl = {1171.11027}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.633/} }
TY - JOUR AU - Koopa Tak-Lun Koo AU - William Stein AU - Gabor Wiese TI - On the generation of the coefficient field of a newform by a single Hecke eigenvalue JO - Journal de théorie des nombres de Bordeaux PY - 2008 SP - 373 EP - 384 VL - 20 IS - 2 PB - Université Bordeaux 1 UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.633/ DO - 10.5802/jtnb.633 LA - en ID - JTNB_2008__20_2_373_0 ER -
%0 Journal Article %A Koopa Tak-Lun Koo %A William Stein %A Gabor Wiese %T On the generation of the coefficient field of a newform by a single Hecke eigenvalue %J Journal de théorie des nombres de Bordeaux %D 2008 %P 373-384 %V 20 %N 2 %I Université Bordeaux 1 %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.633/ %R 10.5802/jtnb.633 %G en %F JTNB_2008__20_2_373_0
Koopa Tak-Lun Koo; William Stein; Gabor Wiese. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.633/
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