I need examples of Jacobi forms for full congruence subgroups $\Gamma(N) $ of $SL(2,Z)$. As a particular case, take the theta function
$\theta_{1,1}(t,z) := \sum_{n\in\mathbb{Z}} exp(\pi it(n + 1/2)^2 + 2\pi i(z + 1/2)(n + 1/2))$.
I know that it is true that $\theta_{1,1}^8 \in J_{4,4}(4)$ where $J_{m,k}(N)$ denotes the space of Jacobi forms of index m and weight k with respect to the congruence group $\Gamma(N)$. Is it true that $\theta_{1,1}^{4N} \in J_{2N,2N}(2N)$? The only think that I don't know how to prove is the property of the Fourier expansion at the cusps, i.e. I need to show that
$\theta_{1,1}^{4N} =\sum_{n,r \text{ with } 4n - r^2>0}c(n,r)q^n\zeta^r$ where $q = e^{\pi i t / N}$ and $\zeta = e^{2\pi i z}$.
Can anyone help?