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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?

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Hint: Say $\gamma_1=\begin{pmatrix}a_1&b_1\\c_1&d_1\end{pmatrix}, \gamma_2=\begin{pmatrix}a_2&b_2\\c_2&d_2\end{pmatrix}, \gamma_1\gamma_2= \begin{pmatrix}a_3&b_3\\c_3&d_3\end{pmatrix}$, What you need to prove is that $[\gamma_1]_2({2\pi ic_2\over c_2\tau+d_2})+{2\pi ic_1\over c_1\tau+d_1}={2\pi c_3\over c_3\tau+d_3}$.

Expand the equation and verify it yourself, don't forget $\gamma\in SL_2$, that is $ad-bc=1$.

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