I have to compute a few terms of the normalized Eisenstein series: $\xi = e^{2 \pi i \tau}$, where $\tau$ belongs to the upper-half plane. In particular, I have to show that:
$$ E_{4} (\tau) =1+240(\xi + ... \xi^{2} + ... \xi^{3}..) = \frac{s_{4}(1,\tau)}{2 \zeta(4)}$$
$$ E_{8} (\tau) =1+(\xi + ... \xi^{2} + ... \xi^{3}..) = \frac{s_{8}(1,\tau)}{2 \zeta(8)}$ $$
then it asks me to conjecture some relation between these two functions("modular forms").
I'm not sure what the $s_{n}(1,\tau)$ functions represent in this case, but the $E_{n}(\tau)$ function comes from the alternate notation when dealing with the $q$-expansion of the Eisenstein series. At least, that is what I understood from this wikipedia link:
https://en.wikipedia.org/wiki/Eisenstein_series
My lecturer has not clarified any of these issues and it is not even covered in the textbook that I'm using. Hence, I would very much appreciate some help in trying to figure out this problem.
