A new look at Hecke's indefinite theta series

TL;DR

This paper introduces a new perspective on Hecke's indefinite theta series by exploring a family of q-series that generate weight 1 modular forms from indefinite quadratic forms and analyzing their linear relations.

Contribution

It presents a novel family of q-series related to indefinite quadratic forms and investigates their linear relations, enriching the understanding of Hecke's indefinite theta series.

Findings

Identified a new family of q-series generating weight 1 modular forms.

Analyzed linear relations among these series.

Enhanced understanding of the structure of Hecke's indefinite theta series.

Abstract

We describe a family of $q$-series generating the space of weight 1 modular forms coming from indefinite binary quadratic forms and study linear relations between these series.