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.