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Based on the notion of strong modular form developed in Part I, we propose tostructure the family of cuspidal modular form spaces$(S_{2k}(\Gamma_0(N)))_{k\in \mathbb{N}^*}$ and to determine bases for each ofthese spaces, once known bases for the first values of $k$.  We then apply these theoretical results to explicitly determine bases forspace families $(S_{2k}(\Gamma_0(N)))_{k\in \mathbb{N}^*}$ when $1\leq N \leq10$.

Structure and bases of modular space sequences  $(M_{2k}(\Gamma_0(N)))_{k\in \mathbb{N}^*}$ and $(S_{2k}(\Gamma_0(N)))_{k\in  \mathbb{N}^*}$. Part III: Cuspidal spaces