Dimension sequences of modular forms

Abstract For N 1 N ≥ 1, let S₂^ { new } (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, S₂^ { new } (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.