The following is from the book Modular Forms by W Stein:
I don't understand the whole sentence from "Note that the set ..." : For example,
1- How is the proving bijection?
2- What does $L=\mathbb{Z}.(a,b) + \mathbb{Z}.(0,d) $ means, i.e. how $.$ is defined? and What does $+$ mean (is it $\oplus$)?
3- How $L$ it is a subgroup of $\mathbb{Z^2}$ of index $n$?