Dimensions of spaces of modular forms

We prove a conjecture of Ross concerning the value distribution of $\dim S_2^{\mathrm{n}\mathrm{e}\mathrm{w}}({\mathrm{\Gamma }}_0(N))$ for $N\in \mathbb{N}$, as well as analogous results for general weight $k\in 2\mathbb{N}$ and the full and twist-minimal spaces $S_k({\mathrm{\Gamma }}_0(N))$, $S_k^{\mathrm{m}\mathrm{i}\mathrm{n}}({\mathrm{\Gamma }}_0(N))$.