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As part of showing that ##E^*_{2}(-1/t)=t^{2}E^*_{2}(t)##

where ##E^*_{2}(t)= - \frac{3}{\pi Im (t) } + E_{2}(t) ##

And since I have that ##t^{-2}E_{2}(-1/t)=E_{2}(t)+\frac{12}{2\pi i t} ##

I conclude that I need to show that ##\frac{-1}{Im(-1/t)}+\frac{2t}{i} = \frac{-t^{2}}{Im(t)} ## [1]

where ##t## inside the upper half plane

Im=imaginary

I also have the identification ##Im(\gamma.t)=\frac{Im(t)}{|ct+d | ^{2}}## [2],

where ##\gamma## is inside ##SL_{2}(Z)##, the modular group of 2x2 matrices with integer numbers and determinant 1, (apologies I'm unsure how you do a matrix in latex), with components ##\gamma_{11}=a,\gamma_{12}=b, \gamma_{21}=c, \gamma_{22}=d ## .

So this is part of a question where I am showing that ## E_{2}(t)^*## is weakly modular via showing that for the generators ##T## and ##S## the relevant identifications hold,##S## and ##T## inside ##SL_{2}(Z)##, so here, the one for ##S## being that :

##f(-1/t)=t^{k}f(t)##, ##S## is the matrix with components ##a=0,b=-1,c=1,d=0##

Now I am looking at [2] and, using ##S## as ##\gamma## that

##Im(S.t)=\frac{Im(t)}{t^{2}}##, but we also know ##S.t=-1/t## and therefore I have ##Im(-1/t)=\frac{Im(t)}{t^{2}}##, and so I have no idea how I'm going to get a ##\frac{-2t}{i}## term in [1]

Many thanks in advance.