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Apéry-like numbers and families of newforms with complex multiplication

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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.

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Acknowledgements

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

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  1. Department of Mathematics & Statistics, Texas Tech University, Lubbock, TX, 79410-1042, USA

    Alexis Gomez, Dermot McCarthy & Dylan Young

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  1. Alexis Gomez
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  2. Dermot McCarthy
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  3. Dylan Young
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Correspondence to Dermot McCarthy.

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Gomez, A., McCarthy, D. & Young, D. Apéry-like numbers and families of newforms with complex multiplication. Res. number theory 5, 5 (2019). https://doi.org/10.1007/s40993-018-0145-7

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