Abstract Algebra- Cosets in $\mathbb{Z}^{2}$

Question

We know that $\mathbb{Z}^{2}$ is a group. Let $2\mathbb{Z}^{2} = \lbrace (2a,2b):a,b \in\mathbb{Z} \rbrace$. $2\mathbb{Z}^{2}$ is the set of vectors in $\mathbb{Z}^{2}$ with both coordinates even. What are the four cosets of $2\mathbb{Z}^{2}$ in $\mathbb{Z}^{2}$?

Answer 1

$$1-(2n+1,2n+1)$$ $$2-(2n,2n+1)$$ $$3-(2n,2n)$$ $$4-(2n+1,2n)$$

I hope you get the idea. It is about odd and even coordinates of $Z^{2}$