Let $\mathcal{S}_k(\Gamma_0(N),\chi)$ denote the space of cuspforms with Dirichlet character $\chi$ and modular subgroup $\Gamma_0(N)$. We characterize the newspace $\mathcal{S}_k^{new}(\Gamma_0(N),\chi)$ as the intersection of eigenspaces of a particular family of Hecke operators generalizing the work of Baruch-Purkait [2015] to forms with non-trivial character. To achieve this, we explicitly describe the Hecke algebra of locally constant compactly supported functions $\mathcal{H}(GL_2(\mathbb{Z}_p)//K_0(p^n),\chi_p)$ where $\chi_{p}$ is a $p$-adic character and $K_0(p^n)$ the Iwahori subgroup of level $n$. We then use this Hecke algebra to describe the irreducible representations of $GL_2(\mathbb{Z}_p)$ that contain a level $n$ fixed vector and identify the new-vector. Finally we de-adelize the above $p$-adic Hecke algebra relations into relations of classical Hecke operators. This approach displays how we can obtain results on the newspace $\mathcal{S}_k^{new}(\Gamma_0(N),\chi)$ from local results.

Bio: Markos Karameris is a third year PhD candidate at the Technion - Israel Institute of Technology under the supervision of Moshe Baruch. His research focuses on automorphic forms and representations with an emphasis on local Hecke Algebras.