Holomorphic Eisenstein modular form (awaiting review)

Let $k$ be a positive integer and let $\Gamma$ be a finite index subgroup of the modular group $\SL(2,\Z)$.

the Eisenstein subspace $E_k(\Gamma)$ is the orthogonal complement in $M_k(\Gamma)$ to the subspace $S_k(\Gamma)$ under the Petersson inner product.

An Eisenstein form of weight $k$ on $\Gamma$ is a modular form $f\in E_k(\Gamma)$. For each Dirichlet character $\chi$ of modulus $N$ the Eisenstein forms in $M_k(N,\chi)$ form a subspace $E_k(N,\chi)$; these are the Eisenstein forms of weight $k$, level $N$, and character $\chi$.

The space $E_k(N, \chi)$ is spanned by the $E_k^{\chi_1, \chi_2}(d \tau)$ where $\chi_1 \chi_2 = \chi$ and $d N_1 N_2 \mid N$, unless $k = 2$ and $\chi = 1$, in which case $E_2^{1,1}(d \tau)$ is not holomorphic, and is replaced by $E_2^{1,1}(\tau) - d E_2^{1,1}(d \tau)$.