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On multiplicativity of Fourier coefficients at cusps other than infinity


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

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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).

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