On page 17 of https://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-74119-0_1/fulltext.pdf, we see a remark where the author mentioned that If $\chi$ is a non-trivial Dirichlet character then there is an Eisenstein series having the Fourier expansion for $$\mathbb{G}_{k,\chi}(z)=c_{k}(\chi)+\sum_{n=1}^{\infty}\Big(\sum_{d|n}\chi(d)d^{k-1}\Big)q^n$$ which satisfies the modularity $$\mathbb{G}_{k,\chi}\Big(\frac{az+b}{cd+d}\Big)=\chi(a)(cz+d)^{k}\mathbb{G}_{k,\chi}(z).$$ When the character $\chi$ is primitive but not-trivial satisfying $\chi(-1)=(-1)^k$, I found no problem with it except that I think one of the "${\chi}$" has lost (typo) a "bar". For example I found it more correct if we write $$\mathbb{G}_{k,\chi}(z)=c_{k}(\chi)+\sum_{n=1}^{\infty}\Big(\sum_{d|n}\overline{\chi}(d)d^{k-1}\Big)q^n.$$ Maybe the book is correct and I am wrong or maybe vice versa.
When $\chi$ is non-primitive satisfying $\chi(-1)=(-1)^k$, my problem is that I have no clue about how to confirm why the same formula for the Eisenstein series can hold (suppose that the book is correct). I know that a combination of "old forms" like $$\sum_{d|\frac{m}{m^\ast}}\frac{a_d}{\sum_{d|\frac{m}{m^\ast}}a_d}f_d(z)$$ satisfies the same modularity condition, where $m$ is the modulus of $\chi$, $\chi^{\ast}$ the primitive character which induces $\chi$ and $m^{\ast}$ its modulus. And $a_d=d^k\mu(m/dm^{\ast})\chi^{\ast}(m/dm^{*})$, $f_d=\mathbb{G}_{k,\chi^{\ast}}(dz)$. But it is for sure not exactly what is written in the book as $\mathbb{G}_{k,\chi}(z)$, hence the book gives a "new" family of Eisenstein series (which might not be a linear combination of "old forms"), about which I would like to know how to compute and also a little bit more details of the its background. Also I would like to know how to prove that $f_d(z)$ in the above formula are linearly independent over $\mathbb{C}$.


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