  Consider $N \geq 1$, $k \geq 2$, and $\chi$ a Dirichlet character modulo $N$
such that $\chi(-1) = (-1)^k$. For any bound $B$, one can show that $\dim
S_k(\Gamma_0(N),\chi) \le B$ for only finitely many triples $(N,k,\chi)$. It
turns out that this property does not extend to the newspace; there exists an
infinite family of triples $(N,k,\chi)$ for which $\dim
S_k^{\text{new}}(\Gamma_0(N),\chi) = 0$. However, we classify this case
entirely. We also show that excluding the infinite family for which $\dim
S_k^{\text{new}}(\Gamma_0(N),\chi) = 0$, $\dim
S_k^{\text{new}}(\Gamma_0(N),\chi) \leq B$ for only finitely many triples
$(N,k,\chi)$. In order to show these results, we derive an explicit dimension
formula for the newspace $S_k^{\text{new}}(\Gamma_0(N),\chi)$. We also use this
explicit dimension formula to prove a character equidistribution property and
disprove a conjecture from Greg Martin that $\dim
S_2^{\text{new}}(\Gamma_0(N))$ takes on all possible non-negative integers.
