Newspaces with Nebentypus: An Explicit Dimension Formula and Classification of Trivial Newspaces

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