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On Decomposition of $θ_2^{2n}(τ)$ as the Sum of Lambert Series and Cusp forms

Based on the values of the Weierstrass elliptic function $\wp(z|τ)$ at $z=πτ/2$, $(π+πτ)/{2}, (π+πτ)/{4},(π+2πτ)/{4}$ and the theory of modular forms on the arithmetic group $Γ_0(2)$, we decompose $θ_2^{2n}(τ)$ as sum of Eisenstein series and a cusp forms. Using the recurrence relation of $\wp^{(2n)}(z|τ)$, we provide an algorithm to determine the exact form of these cusp forms.