Newspaces with nebentypus: An explicit dimension formula and classification of trivial newspaces

Consider $N\ge 1$, $k\ge 2$, and χ a Dirichlet character modulo N such that $\chi (-1)={(-1)}^k$. For any bound B, one can show that $\dim S_k\left({\Gamma }_0\right(N),\chi )\le B$ 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 $\dim S_k^{\text{new}}\left({\Gamma }_0\right(N),\chi )=0$. However, we classify this case entirely. We also show that excluding the infinite family for which $\dim S_k^{\text{new}}\left({\Gamma }_0\right(N),\chi )=0$, $\dim S_k^{\text{new}}\left({\Gamma }_0\right(N),\chi )\le B$ for only finitely many triples $(N,k,\chi )$. In order to show these results, we derive an explicit dimension formula for the newspace $S_k^{\text{new}}\left({\Gamma }_0\right(N),\chi )$. We also use this explicit dimension formula to prove a character equidistribution property and disprove a conjecture from Greg Martin that $\dim S_2^{\text{new}}\left({\Gamma }_0\right(N\left)\right)$ takes on all possible non-negative integers.