Dimension sequences of modular forms

For $N\ge 1$, let $S_2^{\text{new}}(N)$ denote the newspace of cuspidal modular forms of weight $2$ and level $N$. In 2004, Greg Martin conjectured that as a sequence in $N$, $\dim S_2^{\text{new}}(N)$ takes on all possible natural numbers. In this paper, we investigate several generalizations and variations of this type of problem. In each case, we provide a complete characterization of when such a property holds.