I'm currently studying the Eisenstein Series and would like to show that said series is holomorphic. In doing so, I want to show that $$\sum_{(n,m) \in \mathbb{Z}^2} \frac{1}{|m^2 + n^2 - mn|^k} = \sum_{(n,m) \in \mathbb{Z}^2} \frac{1}{|m \gamma - n|^k},$$
converges for $k \ge 3$ and where $\gamma = e^{2\pi i/3}.$ I've tried bounding this by numerous things but I'm not getting anywhere.
Any help showing that this converges would be greatly appreciated - thanks!