# Apéry-like numbers and families of newforms with complex multiplication

Dermot McCarthy, Alexis Gomez, Dylan Young

Research output: Contribution to journalArticlepeer-review

## 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{\'e}ry-like sequences.
Original language English 12 pages Research in Number Theory https://doi.org/10.1007/s40993-018-0145-7 Published - Mar 2019