Weierstrass invariant on a particular lattice

Consider the function

\begin{eqnarray*} \wp(z)=\frac{1}{z^2}+\sum_{a\in \Lambda^*}\left( \frac{1}{(z-a)^2}-\frac{1}{a^2}\right) \end{eqnarray*}

where $\Lambda = \mathbb{Z}\bigoplus\omega\mathbb{Z} $ and $\omega$ is the third root of unity in the upper half plane.

I already know what the zeros of $\wp$ are, namely $\frac{\omega}{\omega - 1}$ and $\frac{\omega}{1-\omega}$ . I need to prove that $g_2(\omega)=60\sum_{\lambda\in\Lambda}\frac{1}{\lambda^4}=0$
however it's not quite clear how to proceed, any help is gladly appreciated.

Thanks to the hint of reuns, we know that from the identity $\omega^2+\omega+1=0$ we can deduce $\omega\Lambda=\Lambda$ , then

$$g_2(\omega)=g_2(\Lambda)=g_2(\omega\Lambda)=60\sum_{\lambda\in\Lambda-0}\frac{1}{(\omega\lambda)^4}=60\omega^2\sum_{\lambda\in\Lambda-0}\frac{1}{\lambda^4}=\omega^2g_2(\omega)$$

From this it clearly follows $g_2(\omega)=0$