An Eisenstein series of rational weight

TL;DR

This paper investigates a specific Eisenstein series of rational weight, establishing its modularity and deriving its Fourier expansion, thereby expanding understanding of modular forms with fractional weights and complex multiplier systems.

Contribution

It proves the modularity of a rational weight Eisenstein series with a complex multiplier system and explicitly computes its Fourier expansion.

Findings

Confirmed the Eisenstein series is a modular form.

Derived explicit Fourier expansion of the series.

Enhanced understanding of rational weight modular forms.

Abstract

We study an Eisenstein series of weight $20/9$, level $11$ whose multiplier system is the same as $\eta^{44/9}(\tau)\eta^{-4/9}(11\tau)$. We prove that it is a modular form and obtain its Fourier expansion.