New Eisenstein subspace (awaiting review)

The space $E_k(N,\chi)$ of Eisenstein modular forms of level $N$, weight $k$, and character $\chi$ can be decomposed \[ E_k(N,\chi) = E_k^{\rm old}(N,\chi) \oplus E_k^{\rm new}(N,\chi) \] into old and new subspaces, defined as follows.

If $M$ is a proper divisor of $N$ and $\chi_M$ is a Dirichlet character of modulus $M$ that induces $\chi$, then for all $d \mid (N/M)$, there is a map from $E_k(M,\chi_M) \to E_k(N,\chi)$ via $f(z) \mapsto f(dz)$. The span of the images of all of these maps is the old subspace $E_k^{\rm old}(N,\chi) \subseteq E_k(N,\chi)$.

The new subspace $E_k^{\rm new}(N,\chi)$ is the subspace spanned by the newforms $E_k^{\chi_1, \chi_2}(\tau)$ such that $\chi_1 \chi_2 = \chi$ and $N_1 N_2 = N$, unless $k = 2$ and $\chi = 1$, in which case $E_2^{\rm new}(N) = 0$ when $N$ is not a prime, and when $N = p$ is prime it is spanned by $E_2^{1,1}(\tau) - p E_2^{1,1}(p \tau)$.