Ap\'ery-like numbers and families of newforms with complex multiplication
Alexis Gomez, Dermot McCarthy, Dylan Young

TL;DR
This paper constructs two infinite families of newforms with complex multiplication using Hecke characters, providing explicit formulas for their Fourier coefficients and establishing relations with Apéry-like sequences.
Contribution
It introduces explicit formulas for Fourier coefficients of newforms with CM and relates them to Apéry-like sequences, advancing understanding of their arithmetic properties.
Findings
Explicit formulas for Fourier coefficients of newforms with CM
Relations between Fourier coefficients of different weights
Congruences involving Apéry-like sequences
Abstract
Using Hecke characters, we construct two infinite families of newforms with complex multiplication, one by and the other by . The values of the -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 -th Fourier coefficients of forms of different weights, within each family. We then prove congruence relations between the -th Fourier coefficients of these newforms at all odd weights and values coming from two of Zagier's sporadic Ap\'ery-like sequences.
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