A modularity criterion for Klein forms, with an application to modular forms of level $13$

TL;DR

This paper establishes a modularity criterion for Klein forms on certain congruence subgroups and applies it to explicitly construct a basis for modular forms of level 13 and weight 2, revealing new properties of theta function coefficients.

Contribution

It introduces a new modularity criterion for Klein forms on $\Gamma_1(N)$ and constructs a basis for modular forms of level 13, weight 2, using this criterion.

Findings

Constructed a basis for modular forms of level 13, weight 2.

Discovered an interesting property of theta function coefficients.

Proved a property conditionally using Hecke operators.

Abstract

We find some modularity criterion for a product of Klein forms of the congruence subgroup $\Gamma_1(N)$ and, as its application, construct a basis of the space of modular forms for $\Gamma_1(13)$ of weight $2$. In the process we face with an interesting property about the coefficients of certain theta function from a quadratic form and prove it conditionally by applying Hecke operators.