Apéry-like numbers and families of newforms with complex multiplication

Abstract

Using Hecke characters, we construct two infinite families of newforms with complex multiplication, one by Q( √ −3) and the other by Q( √ −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.

Generalized Paley graphs and their complete subgraphs of orders three and four

“…When p ≡ 1 (mod 4), γ 1 (p) = 2x [18,30]. When p ≡ 1 (mod 8), γ 3 (p) = 2u 2 − 4v 2 = 4u 2 − 2p [9]. Further, numerical evidence [17] suggests that γ 2 (p) ?…”

“…= −2u, when p ≡ 1 (mod 8). It should be a relatively straightforward exercise to establish this relation using Hecke characters and a similar construction to that in [9]. If p ≡ 1 (mod 4), then [20] In general, establishing relations between finite field hypergeometric functions and coefficients of non-CM modular forms is not a straightforward exercise.…”