Structure and bases of modular space sequences $(M_{2k}(Γ_0(N)))_{k\in \mathbb{N}^*}$ and $(S_{2k}(Γ_0(N)))_{k\in \mathbb{N}^*}$. Part II: a modular butterfly hunt

Abstract: 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 $(M_{2k}(\Gamma_0(N)))_{k\in \mathbb{N}^*}$ 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 $1\leq N \leq 10$.