2
$\begingroup$

Where $E_{2,3}(z) = E_2(z) - 3E_2(3z)$ and $E_2$ is the Eisenstein series of weight 2 and level one.

Now I've shown that the above modular form is in fact in $M_k(\Gamma_0(3))$ so now I just need to show that the dimension of this space is also 1.

My idea was to use the valence formula $$ \sum_{z \in \Gamma \backslash \mathcal{H}} \frac{v_z(f)}{n_\Gamma(z)} + \sum_{c \in C(\Gamma)} v_c(f) = \frac{k\, [PSL_2(\mathbf{Z}) : \overline{\Gamma}]}{12}$$

We know the only cusps of $\Gamma_0(3)$ are $0, \infty$ so the second sum can be computed. I think my main problem comes from a lack of understanding of what is going on in the first sum.

I'd appreciate any help or even a link to a similar example or discussion.

  • 0
    I think the best way to figure out what is going on in that sum is direct computation of a few examples. Try calculating it for $\Delta$ in $M_{12}(\Gamma_0(3))$ first. Once you figure out the sum a bit more, I'll give you a hint to do this is a faster way. (I think it's faster anyway) – 2017-03-16

0 Answers 0