TY - JOUR

T1 - Apéry-like numbers and families of newforms with complex multiplication

AU - McCarthy, Dermot

AU - Gomez, Alexis

AU - Young, Dylan

PY - 2019/3

Y1 - 2019/3

N2 - 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.

AB - 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.

U2 - 10.1007/s40993-018-0145-7

DO - 10.1007/s40993-018-0145-7

M3 - Article

SP - 12 pages

JO - Research in Number Theory

JF - Research in Number Theory

ER -