An upper bound on the least prime for certain modular congruences
Volume 216 / 2024
Acta Arithmetica 216 (2024), 197-212
MSC: Primary 11F33; Secondary 11F80
DOI: 10.4064/aa230713-25-5
Published online: 18 November 2024
Abstract
Let $l\geq 5$ be a prime and $m$ be a positive integer. Recently Ahlgren, Allen and Tang proved several congruences for newforms, such as $f\vert_{T(Q)} \equiv f\ ({\rm mod}\ l^m)$ for all $f\in S_{l-3}^{{\rm new}}(6)$, for a set of primes $Q \equiv 1\ ({\rm mod}\ l^m)$ of positive density. We obtain a bound on the first such prime. As an application, we obtain an upper bound on primes $Q$ which satisfy certain congruences for the partition function.
