Newspaces with Nebentypus: An Explicit Dimension Formula, Classification of Trivial Newspaces, and Character Equidistribution Property

Consider N ≥ 1, k ≥ 2, and χ a Dirichlet character modulo N such that χ(-1) = (-1)ᵏ. For any bound B, one can show that Sₖ(Γ₀(N),χ) ≤ B for only finitely many triples (N,k,χ). It turns out that this property does not extend to the newspace; there exists an infinite family of triples (N,k,χ) for which Sₖⁿᵉʷ(Γ₀(N),χ) = 0. However, we classify this case entirely. We also show that excluding the infinite family for which Sₖⁿᵉʷ(Γ₀(N),χ) = 0, Sₖⁿᵉʷ(Γ₀(N),χ) ≤ B for only finitely many triples (N,k,χ). In order to show these results, we derive an explicit dimension formula for the newspace Sₖⁿᵉʷ(Γ₀(N),χ). We also use this explicit dimension formula to prove a character equidistribution property and disprove a conjecture from Greg Martin that S₂ⁿᵉʷ(Γ₀(N)) takes on all possible non-negative integers.