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 III: Cuspidal spaces

Based on the notion of strong modular form developed in Part I, we propose to structure the family of cuspidal modular form spaces $(S_{2k}({\mathrm{\Gamma }}_0(N)){)}_{k\in {\mathbb{N}}^{\ast }}$ and to determine bases for each of these spaces, once known bases for the first values of $k$. We then apply these theoretical results to explicitly determine bases for space families $(S_{2k}({\mathrm{\Gamma }}_0(N)){)}_{k\in {\mathbb{N}}^{\ast }}$ when $1\leq N \leq 10$.