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.