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, $$ E_2 = \dfrac{\eta(2z)^{20}}{\eta(z)^8\eta(4z)^8} + \dfrac{\eta(4z)^8}{\eta(2z)^4} $$ (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 $\Gamma_0(4)$?

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 $\Gamma_0(4)$?