Finding complete residue systems in Gaussian integers

Question

Let $\mathbb{Z}[i]$ be the ring of Gaussian integers. A set $C\subset \mathbb{Z}[i]$ is called a complete residue system (CRS) modulo $\alpha$ if for every Gaussian integer $\pi$ there exists a unique $\rho \in C$ such that $\alpha$ divides $\pi - \rho.$ I intuitively see that one such system is the set of point enclosed by $\alpha, i \alpha$ , How do I exhibit explicitly such a set, say for $\alpha = 2 + 2i$, and prove that it is a CRS? In addition, how do I show that there are $N(\alpha)$ elements in this residue system? (with $N$ the usual norm on complex numbers.)