Relations between $M_k(\Gamma_0(N))$ and $M_k(\Gamma_0(L,M))$, and Eisenstein Series. Consider the following congruence subgroups
$$\Gamma_0(N):=\Big\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix}\in\text{SL}_2(\mathbb{Z})\; \Big| \; c\equiv0\text{ (mod }N)\Big\}$$
and
$$\Gamma_0(L,M):=\Big\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix}\in\text{SL}_2(\mathbb{Z})\; \Big| \; c\equiv0\text{ (mod }L)\;\text{ and } 
  \;b\equiv0\text{ (mod }M) \Big\}.$$
Is there some sort of relationship between modular forms over these groups? Say, for example, if $N=LM$ or $N=\text{lcm}(N,M)$?
I am asking this because I was reading Miyake's book (titled "Modular Forms"), at chapter 7, for two characters $\chi$ and $\psi$ of modulus $L$ and $M$, he defines the Eisenstein Series $E_k(z,s,\chi,\psi)$. He then proves (for $s=0$ and under some conditions), that they are modular over $\Gamma_0(L,M)$ with nebentypus $\chi\bar{\psi}$.
However, everywhere else that I see these Eisensteins series they are understood as being modular over $\Gamma_0(LM)$. For example https://wstein.org/books/modform/modform/eisenstein.html (see the comments after equation (4)).
I am confused. Are these the same series, or do they require some slight modification to be modular over $\Gamma_0(LM)$?
 A: It's not true that there is a convenient relationship between $\Gamma_0(L, M)$ and $\Gamma_0(N)$. But there is a relationship between the Eisenstein series defined on them. The Eisenstein series in Stein and the Eisenstein series in Miyake aren't quite the same Eisenstein series.
Stein's Eisenstein series appears earlier in Miyake's book. It's the one described in Theorem 4.7.1 (pg 177) of Miyake, which is shown to be modular on $\Gamma_0(LM)$, where $L$ and $M$ are the conductors of the Dirichlet characters (where I use the notation from chapter 7 and your question, which differs from the notation in the book in chapter 4).
Below Theorem 7.1.3 of Miyake, in equation 7.1.13 (pg 271), Miyake states that one gets the Stein-type Eisenstein series via
$$ E_k(Mz; \chi, \psi) = c E_{\mathrm{Stein}}(z; \chi, \overline{\psi}) $$
for a constant $c = c(k, \psi, M)$. Here, $E_{\mathrm{Stein}}$ is my notation for the Stein-type Eisenstein series.
Thus the conversion is to consider $E_k(Mz; \chi, \psi)$, which is modular on $\Gamma_0(LM)$. Stein translates results from Miyake through this identity for his book.
Additional note: a typo in Stein
There is a typo that make this comparison more confusing. Equation 4 in https://wstein.org/books/modform/modform/eisenstein.html (or equivalently equation 5.3.1 in the book version), $\psi(n)$ should be replaced by $\overline{\psi}(n)$. And in Theorem 5.9, the condition should either read
$\chi = \varepsilon \psi$ or $\chi \overline{\psi} = \varepsilon$.
In fact, this was the source of a problem in sagemath for a bit. Here is the source of that trac ticket
