Exercise 12 in Ch. 1 of Apostol's Modular Functions and Dirichlet Series - How to See The Vanishing of Eisenstein Series

Question

The mentioned problem has you prove $$G_{2k}(\frac{-1}{\tau}) = \tau^{2k}G_{2k}(\tau).$$ From there, it asks that you deduce the following, $$G_{2k}(e^{2 \pi {i} / 3})=0$$ if $k \neq 0 \mod 3$. I see that $$G_{2k}(e^{2 \pi {i} / 3})=(-e^{-2 \pi {i} / 3})^{2k}G_{2k}(-e^{-2 \pi {i} /3})$$ But I am not sure how to see that the above is $0$. Any help is appreciated.

Answer 1

I think you also want to use $G_{2k}(\tau+1)=G_{2k}(\tau)$. Notice that if $\tau=e^{2\pi i/3}$, then $-1/\tau=\tau-1$, and it falls apart.