Dimension sequences of modular forms

For N ≥ 1 N ≥ 1 , let S₂ⁿᵉʷ(N) S 2 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 , S₂ⁿᵉʷ(N) dim S 2 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.