On Decomposition of ${\theta }_2^{2n}(\tau )$ as the Sum of Lambert Series and Cusp forms

Based on the values of the Weierstrass elliptic function $\mathrm{\wp }(z\mathrm{\mid }\tau )$ at $z=\pi \tau \mathrm{/}2$, $(\pi +\pi \tau )\mathrm{/}2,(\pi +\pi \tau )\mathrm{/}4,(\pi +2\pi \tau )\mathrm{/}4$ and the theory of modular forms on the arithmetic group ${\mathrm{\Gamma }}_0(2)$, we decompose ${\theta }_2^{2n}(\tau )$ as sum of Eisenstein series and a cusp forms. Using the recurrence relation of ${\mathrm{\wp }}^{(2n)}(z\mathrm{\mid }\tau )$, we provide an algorithm to determine the exact form of these cusp forms.