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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: winfried@mathi.uni-heidelberg.de)
YUK-KAM LAU
Affiliation:
Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong (email: yklau@maths.hku.hk)
IGOR E. SHPARLINSKI*
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia (email: igor@ics.mq.edu.au)
*
✉For correspondence; e-mail: igor@ics.mq.edu.au
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Abstract

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We show 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.

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