Modular Forms, Eisenstein series, show it transforms with modular of weight 2

[binbagsss]

Homework Statement

I need to show that

transforms with modular of weight $2$ for $SL_2(Z)$

We have the theorem that it is sufficient to check the generators S and T

We have that E_2 is (whilst holomorphic) fails to transform with modular weight $2$ as it has this extra term when checking for $S$:

where transforming with weight $2$ means (for $S$):

$f(S.\tau)=\tau^{2}f(\tau)$Therefore we expect a cancellation from the $Im (\tau)$ term

We have

MY QUESTION
- I follow this solution and this is fine
- I guess I am doing something pretty stupid, but we have the following formula

for $G=SL_2(R)$ and since $Z \in R$ we have $SL_2(Z) \in SL_2(R)$, so the above also holds for $SL_2(Z)$ and so this gives:

$Im(S.\tau)=\frac{Im(\tau)}{\tau^2}$- so $1/Im(\tau)$ transforms with modular weight $2$ itself, and so is not cancelling
- I perhaps thought there may have been an issue that $\infty$ is not taken into account, but it also says the action on $G$ extends to $\infty$ , as I've said I follow the above solution, but have no idea what is wrong with using this formula.

Many thanks in advance

Homework Equations

The Attempt at a Solution

 

[Nino MMP]

We have that E_2 is (whilst holomorphic) fails to transform with modular weight $2$ as it has this extra term when checking for $S$: where transforming with weight $2$ means (for $S$):$f(S.\tau)=\tau^{2}f(\tau)$Therefore we expect a cancellation from the $Im (\tau)$ term which isn't present.