Let $q=e^{2\pi i\tau}$ and 
$$E_2(\tau) = 1 - 24 \sum_{n=1}^\infty\frac{nq^n}{1-q^n}$$ be the Eisenstein Series of weight $2$
and let $E_2^*(\tau) = E_2(\tau) - \frac{3}{\pi\cdot Im(\tau)}$ be the corresponding *almost holomorphic modular form*. Then let
$$\eta(\tau)=e^{\pi i\tau /12}\cdot\prod_{n=1}^\infty(1-q^n)$$
be the Dedekind $\eta$-Function.

**Question: How can I prove the following statement from [here][1]:**

> Let $\tau$ be any Complex Multiplication point. By basic theorems of complex multiplication, if you choose a suitable period
$\omega(\tau)$, $E_4(\tau)/\omega(\tau)^4$, $E_6(\tau)/\omega(\tau)^6$,
and $\sqrt{D}E_2^*(\tau)/\omega(\tau)^2$ (with $E_2^*(\tau)=E_2(\tau)-3/(\pi \cdot Im(\tau))$ and $D$ the discriminant of $\tau$) will be algebraic numbers of known degree, and if you choose
$\omega(\tau)=\eta(\tau)^2$, they will even be algebraic integers.


**Partial Solution for $E_2^*$:**

Francois Brunault pointed out that this statement can be found in Prop. 5.10.6 on p. 202 of Cohen/Strömberg's book [Modular Forms: A Classical Approach][2].
Unfortunately, the proof of this proposition starts with "we only prove algebraicity, not the integrality". ***Who can help with proving the integrality? Since I am no expert in complex multiplication, I am looking for a rather detailed answer, or for a reference.***

**Complete Solution for $E_4$ and $E_6$:**

The statement that $\frac{E_4}{\eta^8}$ is an algebraic integer follows from
$$\left(\frac{E_4}{\eta^8}\right)^3 = j(\tau)$$
and the statement that $\frac{E_6}{\eta^{12}}$ is an algebraic integer follows from
$$\left(\frac{E_6}{\eta^{12}}\right)^2 = j(\tau)-1728$$
with the absolute invariant $j(\tau)=\frac{1728E_4^3}{E_4^3-E_6^2}=\frac{E_4^3}{\eta^{24}}$ which is an algebraic integer (see Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Theorem 6.1, p. 140).


**EDIT: Complete Solution for $E_2^*$:**

The answer of Michael Griffin (see below) can now be found in more details in the appendix of [this][3] arXiv-preprint.

  [1]: https://mathoverflow.net/questions/300640/how-to-compute-coefficients-in-chudnovskys-formula/300656#300656
  [2]: https://books.google.fr/books?id=8R0vDwAAQBAJ&pg=PA202#v=onepage&q&f=false
  [3]: https://arxiv.org/abs/1809.00533