Structure and bases of modular space sequences $(M_{2k}({\mathrm{\Gamma }}_0(N)){)}_{k\in {\mathbb{N}}^{\ast }}$ and $(S_{2k}({\mathrm{\Gamma }}_0(N)){)}_{k\in {\mathbb{N}}^{\ast }}$. Part II: a modular butterfly hunt

In the first part of this article, which contains three of them, we have identified the notion of level $N$ strong modular unit. It enabled us to structure the modular forms family $(M_{2k}({\mathrm{\Gamma }}_0(N)){)}_{k\in {\mathbb{N}}^{\ast }}$ and to propose the explicit bases for these spaces. It is in this perspective that we wrote this second part where the structure and explicit bases are proposed when $1\le N\le 10$.