On spaces of modular forms spanned by eta-quotients

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

“…Proof. The relation (13) is clearly deduced from the equalities (11) and (12), then the modular character of (12) will be established in the following paragraph.…”