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On multiplicativity of Fourier coefficients at cusps other than infinity

The Ramanujan Journal volume 34, pages 283–306 (2014)Cite this article

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.

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Acknowledgement

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

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  1. Department of Mathematics, Mailcode 4408, Southern Illinois University, 1245 Lincoln Drive, Carbondale, IL, 62901, USA

    Joseph Hundley

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  1. Joseph Hundley
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Correspondence to Joseph Hundley.

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This paper was written while the author was supported by NSF Grant DMS-1001792.

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Hundley, J. On multiplicativity of Fourier coefficients at cusps other than infinity. Ramanujan J 34, 283–306 (2014). https://doi.org/10.1007/s11139-013-9541-8

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  • DOI: https://doi.org/10.1007/s11139-013-9541-8

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