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The Ramanujan Journal

, Volume 34, Issue 2, pp 283–306 | Cite as

On multiplicativity of Fourier coefficients at cusps other than infinity

  • Joseph HundleyEmail author
Article
  • 147 Downloads

Abstract

This paper treats the problem of determining conditions for the Fourier coefficients of a Maass–Hecke newform at cusps other than infinity to be multiplicative. To be precise, the Fourier coefficients are defined using a choice of matrix in \(\mathit{SL}(2, \mathbb{Z})\) which maps infinity to the cusp in question. Let c and d be the entries in the bottom row of this matrix, and let N be the minimal level. In earlier work with Dorian Goldfeld and Min Lee, we proved that the coefficients will be multiplicative whenever N divides 2cd. This paper proves that they will not be multiplicative unless N divides 576cd. Further, if one assumes that the Hecke eigenvalue vanishes less than half the time, then this number drops to 4cd, and a precise condition governing the case where N divides 4cd and not 2cd is obtained.

Keywords

Fourier coefficients Cusp Maass–Hecke newform Multiplicative relations 

Mathematics Subject Classification

11F06 11F11 11F30 

Notes

Acknowledgement

The author would like to thank Dorian Goldfeld, Min Lee, and Ravi Ragunathan for helpful conversations.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematics, Mailcode 4408Southern Illinois UniversityCarbondaleUSA

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