Eisenstein newform (awaiting review)

An Eisenstein newform is an Eisenstein form $f\in E_k^{\rm new}(N,\chi)$ in the Eisenstein new subspace that is also an eigenform of all Hecke operators, normalized so that the $q$-expansion $f(z)=\sum a_n q^n$, where $q=e^{2\pi i z}$, has coefficient $a_1=1$. The Eisenstein newforms are a basis for the Eisenstein new subspace.