I Am wondering if $f(z)=G_2(z)- \frac{ \pi}{Im(z)}$ is a modular form on $SL_2(\mathbb{Z})$, given that $f: \mathbb{H} \rightarrow \mathbb{C}$.
Here is what I have so far:
Step one:
I tried to show $f(\gamma z)=j(\gamma,z)^2 f(z) \forall\gamma \in SL_2(\mathbb{Z}).$ (Remark: $j(\gamma,z)= cz+d$
Attempt: $G_2(\gamma z)= G_2 \left(\frac{az+b}{cz+d} \right)= \sum_{{m,n \in Z \\ m,n \not= (0,0)}} \frac{1}{(m \frac{(az+b)}{(cz+d)}+n)^2}=(cz+d)^2 \sum_{{m,n \in Z \\ m,n \not= (0,0)}} \frac{1}{z(ma+nc)+mb+nd} = j(\gamma,z)^2 \sum_{{m,n \in Z \\ m,n \not= (0,0)}} \frac{1}{(ze+f)^2} =j(\gamma,z)^2 G_2(z)$
by letting $e=ma+nc$ and $f=mb+nd$
We also know $\frac{\pi}{\Im(\gamma z)}=\frac{\pi |j(\gamma,z)|^2}{\Im(z)}$ Since $j(\gamma,z)^2 \neq |j(\gamma,z)|^2 $ I am unable to conclude that $f(\gamma z)= j(\gamma,z)^2 f(x)$.
Step 2: I want to know if it is meromorphic at $\infty$, i.e. If $f^{\sim}(q)= f\left( \frac{\log q}{2 \pi i}\right)$ can be written as a series with $a_0=0$.
I am unsure how to proceed for this step.