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Apéry-like numbers and families of newforms with complex multiplication

  • Alexis Gomez
  • Dermot McCarthyEmail author
  • Dylan Young
Research
  • 23 Downloads

Abstract

Using Hecke characters, we construct two infinite families of newforms with complex multiplication, one by \({\mathbb {Q}}(\sqrt{-3})\) and the other by \({\mathbb {Q}}(\sqrt{-2})\). The values of the p-th Fourier coefficients of all the forms in each family can be described by a single formula, which we provide explicitly. This allows us to establish a formula relating the p-th Fourier coefficients of forms of different weights, within each family. We then prove congruence relations between the p-th Fourier coefficients of these newforms at all odd weights and values coming from two of Zagier’s sporadic Apéry-like sequences.

Mathematics Subject Classification

Primary: 11F11 11B83 Secondary: 11A07 

Notes

Author's contributions

Acknowledgements

The second author is supported by a grant from the Simons Foundation (353329, Dermot McCarthy).

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Copyright information

© SpringerNature 2018

Authors and Affiliations

  1. 1.Department of Mathematics & StatisticsTexas Tech UniversityLubbockUSA

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