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

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.