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

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## Abstract

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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.

## How to cite

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Koo, 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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1. S. Baba, R. Murty, Irreducibility of Hecke Polynomials. Math. Research Letters 10 (2003), no. 5-6, 709–715. Zbl1162.11335MR2024727
2. D. W. Farmer, K. James, The irreducibility of some level 1 Hecke polynomials. Math. Comp. 71 (2002), no. 239, 1263–1270. Zbl0995.11032MR1898755
3. W. Fulton, J. Harris, Representation Theory, A First Course. Springer, 1991. Zbl0744.22001MR1153249
4. J. Kevin, K. Ono, A note on the Irreducibility of Hecke Polynomials. J. Number Theory 73 (1998), 527–532. Zbl0931.11011MR1658012
5. K. A. Ribet, Twists of modular forms and endomorphisms of abelian varieties. Math. Ann. 253 (1980), no. 1, 43–62. Zbl0421.14008MR594532
6. K. A. Ribet, On l-adic representations attached to modular forms. II. Glasgow Math. J. 27 (1985), 185–194. Zbl0596.10027MR819838
7. J.-P. Serre, Quelques applications du théorème de densité de Chebotarev. Inst. Hautes Etudes Sci. Publ. Math. 54 (1981), 323–401. Zbl0496.12011MR644559
8. W. Stein, Sage Mathematics Software (Version 3.0). The SAGE Group, 2008, http://www.sagemath.org.

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