On Decomposition of $\theta_2^{2n}(\tau)$ as the Sum of Lambert Series and Cusp forms

TL;DR

This paper presents a method to decompose the power of the theta function ^{2n}( au) into Eisenstein series and cusp forms using elliptic functions and modular form theory, providing an algorithm for explicit decomposition.

Contribution

It introduces a novel decomposition of ^{2n}( au) into modular components and an algorithm to explicitly determine cusp forms based on elliptic function values and recurrence relations.

Findings

Decomposition of ^{2n}( au) into Eisenstein series and cusp forms.

An algorithm for explicit determination of cusp forms.

Application of elliptic function values and recurrence relations.

Abstract

Based on the values of the Weierstrass elliptic function $\wp(z|\tau)$ at $z=\pi\tau/2$, $(\pi+\pi\tau)/{2}, (\pi+\pi\tau)/{4},(\pi+2\pi\tau)/{4}$ and the theory of modular forms on the arithmetic group $\Gamma_0(2)$, we decompose $\theta_2^{2n}(\tau)$ as sum of Eisenstein series and a cusp forms. Using the recurrence relation of $\wp^{(2n)}(z|\tau)$, we provide an algorithm to determine the exact form of these cusp forms.