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 $\Gamma\_0(4)$?



Modularity of $E_2$ on congruence subgroups
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.]


