# On a $q$-deformation of modular forms

## TL;DR

This paper introduces a hypergeometric q-deformation of two CM modular forms of weight 3, linking Fourier coefficients, hypergeometric series, and q-congruences to deepen understanding of modular form properties.

## Contribution

It constructs a novel hypergeometric q-deformation of specific CM modular forms and explores associated q-congruences, combining previous insights into a new framework.

## Key findings

- Established a hypergeometric q-deformation for two CM modular forms of weight 3.
- Identified new q-congruences related to the deformed modular forms.
- Connected Fourier coefficients with hypergeometric series at roots of unity.

## Abstract

There are many instances known when the Fourier coefficients of modular forms are congruent to partial sums of hypergeometric series. In our previous work arXiv:1803.01830, such partial sums are related to the radial asymptotics of infinite $q$-hypergeometric sums at roots of unity. Here we combine the two features to construct a hypergeometric $q$-deformation of two CM modular forms of weight 3 and discuss the corresponding $q$-congruences.