Ap\'ery-like numbers and families of newforms with complex multiplication

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 $\mathbb{Q}(\sqrt{-3})$ 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 Ap\'ery-like sequences.