# An Analogue of the Dedekind Eta Function for Hecke Groups $H(\sqrt{D})$

## TL;DR

This paper constructs a new family of holomorphic modular functions for Hecke groups associated with real quadratic fields, extending the classical Dedekind eta function and analyzing their Fourier coefficients.

## Contribution

It introduces an analogue of the Dedekind eta function for Hecke groups $H(\sqrt{D})$, providing new modular functions with detailed growth and sign pattern analysis.

## Key findings

- Holomorphic modular functions vanish at the cusp at infinity.
- Fourier coefficients exhibit specific asymptotic growth patterns.
- Sign patterns of Fourier coefficients are characterized.

## Abstract

Let $D\equiv 1\bmod{4}$ be a fundamental discriminant of a real quadratic field. We construct an analogue of the classical Dedekind eta function for the Hecke group $H(\sqrt{D})$. This gives rise to a new family of holomorphic modular functions for $H(\sqrt{D})$ which vanish at the cusp at $\infty$. We establish results on the asymptotic growth and sign patterns of the Fourier coefficients associated to these modular forms.