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

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 $Γ_0(4)$ for any fixed weight. There are also some new results on $n$-fold covers of modular groups for $n\geq3$. Finally, we provide SageMath programs for verifying the theorems and generating the tables.