I'm new to modular form, reading the book A First Course in Modular Forms
We have the weight 2 Eisenstein series $$ G_2(\tau)=\sum_{c\in\mathbb{Z}}\sum_{d\in\mathbb{Z}_c'}\frac{1}{(c\tau+d)^2} $$
where $\mathbb{Z}_c'=\mathbb{Z}-\{0\}$ when $c=0$ otherwise $\mathbb{Z}$.
Now I am given that $$ (G_2[\gamma]_2)(\tau)=G_2(\tau)-\frac{2\pi ic}{c\tau+d}\text{for }\gamma=\begin{bmatrix}a&b\\c&d\end{bmatrix}\in\text{SL}_2(\mathbb{Z}) $$
And I am asked to prove that if we know that the above formula is correct for two particular matrices $\gamma_1,\gamma_2$,then it is correct for $\gamma_1\gamma_2$.
I try to do as following: $$ \begin{align*} (G_2[\gamma_1\gamma_2]_2)(\tau)&=(G_2[\gamma_1]_2[\gamma_2]_2)(\tau)\text{ by property of the operator}\\ &=(G_2[\gamma_1]_2)(\gamma_2(\tau))\cdot j(\gamma_2,\tau)^{-1}\text{ by the definition} \end{align*} $$
But after substitute $\tau$ by $\gamma_2(\tau)$ in the above given formula, I can not get the desired equation.
Can anyone help?