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Based on the values of the Weierstrass elliptic function $\wp(z|\tau)$ at$z=\pi\tau/2$, $(\pi+\pi\tau)/{2}, (\pi+\pi\tau)/{4},(\pi+2\pi\tau)/{4}$ andthe 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 therecurrence relation of $\wp^{(2n)}(z|\tau)$, we provide an algorithm todetermine the exact form of these cusp forms.

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