Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 1 / a / Solution

The dimension of level-one cusp forms is zero when $k$ is odd or $k<12$. For even $k\ge 12$ it isEquivalently, multiplication by the modular discriminant gives $S_k(\Gamma (1))=\Delta M_{k-12}(\Gamma (1))$, and the displayed formula follows from the standard dimension formula for $M_{k-12}(\Gamma (1))$.