# A ring of symmetric Hermitian modular forms of degree $2$ with integral Fourier coefficients

## TL;DR

This paper characterizes the structure of the ring of symmetric Hermitian modular forms of degree 2 over the integers, providing generators, and establishes Sturm bounds for certain weights and primes.

## Contribution

It explicitly determines the generators of the integral symmetric Hermitian modular forms ring of degree 2 and derives Sturm bounds for specific primes and weights.

## Key findings

- Identified 24 generators for the modular forms ring.
- Established Sturm bounds for weights divisible by 4 at primes 2 and 3.
- Provided structure theorem for the ring over integers.

## Abstract

We determine the structure over $\mathbb{Z}$ of the ring of symmetric Hermitian modular forms with respect to $\mathbb{Q}(\sqrt{-1})$ of degree $2$ (with a character), whose Fourier coefficients are integers. Namely, we give a set of generators consisting of $24$ modular forms. As an application of our structure theorem, we give the Sturm bounds of such the modular forms of weight $k$ with $4\mid k$, in the case $p=2$, $3$. We remark that the bounds for $p\ge 5$ are already known.