The Ramanujan Journal

, Volume 7, Issue 1, pp 269–277

On the Signs of Fourier Coefficients of Cusp Forms

  • Marvin Knopp
  • Winfried Kohnen
  • Wladimir Pribitkin

DOI: 10.1023/A:1026207515396

Cite this article as:
Knopp, M., Kohnen, W. & Pribitkin, W. The Ramanujan Journal (2003) 7: 269. doi:10.1023/A:1026207515396


Let Γ be a discrete subgroup of SL(2, \(\mathbb{R}\)) with a fundamental region of finite hyperbolic volume. (Then, Γ is a finitely generated Fuchsian group of the first kind.) Let\(f(z) = \sum\limits_{n + {\kappa > 0}} {a(n)e^{2\pi i(n + {\kappa })z/{\lambda }} } ,{ }z \in \mathcal{H}.\)be a nontrivial cusp form, with multiplier system, with respect to Γ. Responding to a question of Geoffrey Mason, the authors present simple proofs of the following two results, under natural restrictions upon Γ.

Theorem.If the coefficients a(n) are real for all n, then the sequence {a(n)} has infinitely many changes of sign.

Theorem.Either the sequence {Re a(n)} has infinitely many sign changes or Re a(n) = 0 for all n. The same holds for the sequence {Im a(n)}.

cusp forms Fourier coefficients 

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Marvin Knopp
    • 1
  • Winfried Kohnen
    • 2
  • Wladimir Pribitkin
    • 3
  1. 1.Department of MathematicsTemple UniversityPhiladelphia
  2. 2.Mathematisches Institut, INF 288Universität at HeidelbergHeidelbergGermany
  3. 3.Department of MathematicsHaverford CollegeHaverford

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