Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-18T13:44:36.210Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE NUMBER OF SIGN CHANGES OF HECKE EIGENVALUES OF NEWFORMS

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 August 2008

WINFRIED KOHNEN ,
YUK-KAM LAU  and
IGOR E. SHPARLINSKI
WINFRIED KOHNEN
Affiliation:
Mathematisches Institut, Universität Heidelberg, D-69120 Heidelberg, Germany (email: [email protected])
YUK-KAM LAU
Affiliation:
Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong (email: [email protected])
IGOR E. SHPARLINSKI*
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Weshow that, for every x exceeding some explicit bound depending only on k and N, there are at least C(k,N)x/log 17x positive and negative coefficients a(n) with n≤x in the Fourier expansion of any non-zero cuspidal Hecke eigenform of even integral weight k≥2 and squarefree level N that is a newform, where C(k,N) depends only on k and N. From this we deduce the existence of a sign change in a short interval.

Keywords

Hecke eigenvaluessign changes

MSC classification

Secondary: 11F11: Holomorphic modular forms of integral weight 11F30: Fourier coefficients of automorphic forms
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 85 , Issue 1 , August 2008 , pp. 87 - 94
Copyright
Copyright © 2008 Australian Mathematical Society

Footnotes

The third author was supported in part by ARC grant DP0556431 during the preparation of this paper.

References

[1]Alkan, E. and Zaharescu, A., ‘Nonvanishing of Fourier coefficients of newforms in progressions’, Acta Arith. 116 (2005), 81–98.CrossRefGoogle Scholar
[2]Blomer, V., ‘On cusp forms associated with binary theta series’, Arch. Math. 82 (2004), 140–146.CrossRefGoogle Scholar
[3]Choie, Y. J., Kohnen, W. and Ono, K., ‘Linear relations between modular forms and non-ordinary primes’, Bull. London Math. Soc. 37 (2005), 335–341.CrossRefGoogle Scholar
[4]Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers (Oxford University Press, Oxford, 1979).Google Scholar
[5]Hoffstein, J. and Lockart, P., ‘Coefficients of Maass forms and the Siegel zero’, Ann. of Math. 140 (1994), 161–181 (with an appendix by D. Goldfield, J. Hoffstein and D. Lieman).Google Scholar
[6]Iwaniec, H. and Kowalski, E., Analytic Number Theory (American Mathematical Society, Providence, RI, 2004).Google Scholar
[7]Iwaniec, H., Kohnen, W. and Sengupta, J., ‘The first negative Hecke eigenvalue’, Int. J. Number Theory 3 (2007), 355–363.CrossRefGoogle Scholar
[8]Knopp, M., Kohnen, W. and Pribitkin, W., ‘On the signs of Fourier coefficients of cusp forms’, Ramanujan J. 7 (2003), 269–277.CrossRefGoogle Scholar
[9]Kohnen, W. and Sengupta, J., ‘On the first sign change of Hecke eigenvalues of newforms’, Math. Z. 254 (2006), 173–184.Google Scholar
[10]Murty, R., ‘Oscillations of Fourier coefficients of modular forms’, Math. Ann. 262 (1983), 431–446.CrossRefGoogle Scholar
[11]Ono, K., The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series (American Mathematical Society, Providence, RI, 2004).Google Scholar
[12]Siegel, C. L., ‘Berechnung von Zetafunktionen an ganzzahligen Stellen’, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1969), 87–102.Google Scholar