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

Koopa Tak-Lun Koo^{[1]}; William Stein^{[1]}; Gabor Wiese^{[2]}

- [1] Department of Mathematics University of Washington Box 354350 Seattle, WA 98195 USA
- [2] Institut für Experimentelle Mathematik Universität Duisburg-Essen Ellernstraße 29 45326 Essen Germany

Journal de Théorie des Nombres de Bordeaux (2008)

- Volume: 20, Issue: 2, page 373-384
- ISSN: 1246-7405

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topKoo, Koopa Tak-Lun, Stein, William, and Wiese, Gabor. "On the generation of the coefficient field of a newform by a single Hecke eigenvalue." Journal de Théorie des Nombres de Bordeaux 20.2 (2008): 373-384. <http://eudml.org/doc/10842>.

@article{Koo2008,

abstract = {Let $f$ be a non-CM newform of weight $k \ge 2$. 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 for further investigation.},

affiliation = {Department of Mathematics University of Washington Box 354350 Seattle, WA 98195 USA; Department of Mathematics University of Washington Box 354350 Seattle, WA 98195 USA; Institut für Experimentelle Mathematik Universität Duisburg-Essen Ellernstraße 29 45326 Essen Germany},

author = {Koo, Koopa Tak-Lun, Stein, William, Wiese, Gabor},

journal = {Journal de Théorie des Nombres de Bordeaux},

keywords = {Hecke eigenvalue; newform},

language = {eng},

number = {2},

pages = {373-384},

publisher = {Université Bordeaux 1},

title = {On the generation of the coefficient field of a newform by a single Hecke eigenvalue},

url = {http://eudml.org/doc/10842},

volume = {20},

year = {2008},

}

TY - JOUR

AU - Koo, Koopa Tak-Lun

AU - Stein, William

AU - Wiese, Gabor

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

PB - Université Bordeaux 1

VL - 20

IS - 2

SP - 373

EP - 384

AB - Let $f$ be a non-CM newform of weight $k \ge 2$. 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 for further investigation.

LA - eng

KW - Hecke eigenvalue; newform

UR - http://eudml.org/doc/10842

ER -

## References

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- K. A. Ribet, On l-adic representations attached to modular forms. II. Glasgow Math. J. 27 (1985), 185–194. Zbl0596.10027MR819838
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- W. Stein, Sage Mathematics Software (Version 3.0). The SAGE Group, 2008, http://www.sagemath.org.

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