Modularity of $E_2$ on congruence subgroups So it is well know that the Eisenstein series of weight 2 is not modular on $SL_2(\mathbb{Z})$. 
In this paper of Kilford (http://uk.arxiv.org/PS_cache/math/pdf/0701/0701478v1.pdf) on page 4, he says that $E_2 \in M_2(\Gamma_0(2))$. 
In fact we can obtain the following equality,
$$
E_2 = \dfrac{\eta(2z)^{20}}{\eta(z)^8\eta(4z)^8} + 16\cdot \dfrac{\eta(4z)^8}{\eta(2z)^4}
$$
(there's a typo in the orignal paper which is fixed here) and since the linear combination of eta-quotients on the right hand side is modular on $\Gamma_0(8)$ by a theorem of Ligozat (which can be found on page 2 of that same paper), we would have that $E_2$ is modular on $\Gamma_0(8)$ as well. 
So my question is, do we have modularity for $E_2$ on $\Gamma_0(2)$ as well? I was wondering this since Kilford says that $E_2 \in M_2(\Gamma_0(2))$. If so, why? How about for $\Gamma_0(4)$?
 A: Kilford uses not $E_2$ but something he calls $E_{2, 2}$, which is $$E_{2}(z) - 2E_2(2z) = 1 - 24 \sum_{n \text{ odd}} \sigma_{1}(n) q^n.$$ This is a modular form of weight 2 and level $\Gamma_0(2)$. Similarly $E_{p, 2} = E_2(z) - p E_2(pz)$ is modular of level $\Gamma_0(p)$ for any $p$.
The naive Eisenstein series $$E_{2}(z) = 1 - 24 \sum_{n \in \mathbb{N}} \sigma_{1}(n) q^n$$ is not a modular form of any level. 
[EDIT: My original answer contained the following statement, which is obviously wrong: "this follows from the fact that $E_2(-1/z) - z^2 E_2(z)$ is something like $6/\pi \mathrm{Im}(z)$, while for a modular form it would have to be holomorphic."
A hopefully better statement is: we have $E_2(z) - z^{-2} E_2(-1/z) = 2\pi i / z$, and if $E_2$ were modular (of some level) then $z^{-2} E_2(-1/z)$ would also be modular (of some other level) and hence their difference would be modular (for the intersection of the two level groups). But a non-constant rational function cannot be modular of any level.]
