The Eisenstein series $$ G_{2k} = \sum_{(m,n) \neq (0,0)} \frac{1}{(m + n\tau)^{2k}} $$ are modular forms (if $k>1$) of weight $2k$ and quasi-modular if $k=1$. It is clear that given modular forms $f,g$ of weight $2k$ and $2\ell$ that $f\cdot g$ is a modular form of weight $2(k + \ell)$.

We can also define modular forms of half-integral weight if we are a little more careful. However, the functional equation $$ f\Big(\frac{az + b}{cz+d}\Big) = (cz+d)^{2k}f(z) $$ must be replaced with something more subtle.

In particular, the Dedekind $\eta$-function is a modular form of weight 1/2; it satisfies $$ \eta(z + 1) = e^{\frac{\pi i}{12}}\eta(z) \qquad \eta\Big(-\frac{1}{z}\Big) = \sqrt{-iz}\ \eta(z) $$ Now, it would be nice if $\eta^4$ were a modular form of weight 2; however, an easy check using the above relations shows that this is only the case up to roots of unity, and so $\eta^4$ is not a multiple of $G_2$.

My question is then the following: What is the relation between modular forms of half-integral weight and (quasi-)modular forms of even integer weight? I know that $\eta^{24}$ is an honest modular form of weight 12, so I'm more curious about the general setting, or even what can be said about things like $\eta^4$.

**Edit:** As was pointed out in the comments, $\eta^4$ is a modular form for a congruence subgroup of $SL_2(\mathbb{Z})$, but not for the full modular group. The space of modular forms for the full modular group is generated by $G_4$ and $G_6$ (with $G_2$ thrown in if we are looking at the space of quasi-modular forms); is there a corresponding statement for modular forms on congruence subgroups?