Dimension formulas for modular form spaces of rational weights, the classification of eta-quotient characters and an extension of Martin's theorem

TL;DR

This paper derives explicit dimension formulas for rational-weight modular forms with eta-quotient multipliers, extends Martin's eta-quotient classification, and provides computational tools and tables for these forms.

Contribution

It introduces explicit formulas for rational-weight modular form dimensions, extends the classification of eta-quotients, and offers computational resources and comprehensive tables.

Findings

Explicit dimension formulas for rational-weight modular forms.

Extension of Martin's eta-quotient classification to fractional exponents.

A table of 2277 eta-quotients and classification of multiplier systems.

Abstract

We give an explicit formula for dimensions of spaces of rational-weight modular forms whose multiplier systems are induced by eta-quotients of fractional exponents. As the first application, we give series expressions of Fourier coefficients of the $n$-th root of certain infinite $q$-products. As the second application, we extend Yves Martin's list of multiplicative holomorphic eta-quotients of integral weights by first extending the meaning of multiplicativity, then identifying one-dimensional spaces, and finally applying Wohlfahrt's extension of Hecke operators. A table containing $2277$ of such eta-quotients is presented. As a related result, we completely classify the multiplier systems induced by eta-quotients of integral exponents. For instance, there are totally $384$ such multiplier systems on $\Gamma_0(4)$ for any fixed weight. We also provide SageMath programs on checking the theorems and generating the tables.