Dimension sequences of modular forms

For N ≥ 1, let S₂ⁿᵉʷ(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, S₂ⁿᵉʷ(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.