Congruences via modular forms

TL;DR

This paper proves new congruences for coefficients of modular forms' power series, settling recent conjectures and providing a comprehensive table of related congruences relevant to differential equations and number theory.

Contribution

It establishes two key congruences for modular form coefficients and confirms two recent conjectures, advancing understanding in modular forms and related number theory.

Findings

Proved two new congruences for modular form coefficients.

Settled two conjectures posed by Chan, Cooper, and Sica.

Provided a comprehensive table of related congruences.

Abstract

We prove two congruences for the coefficients of power series expansions in t of modular forms where t is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide a table of congruences for numbers which appear in similar power series expansions and in the study of integral solutions of Apery-like differential equations.