A non-ordinary (prime) note

TL;DR

This paper investigates primes dividing Fourier coefficients of certain newforms, revealing that for some weight 4 newforms, divisibility by a prime p implies stronger divisibility properties, challenging expectations of rarity.

Contribution

It demonstrates that for specific weight 4 newforms, divisibility by a prime p leads to stronger divisibility, highlighting unexpected patterns in Fourier coefficients.

Findings

Identification of newforms with prime divisibility properties

Examples of primes p dividing Fourier coefficients with stronger divisibility

Insight into the rarity and significance of such divisibility patterns

Abstract

Given a newform with the Fourier expansion $\sum_{n=1}^\infty b(n)q^n\in\mathbb Z[[q]]$, a prime $p$ is said to be non-ordinary if $p\mid b(p)$. We exemplify several newforms of weight 4 for which the latter divisibility implies a stronger divisibility - a property that may be thought unlikely to happen too often.