# Coefficients of Modular Forms

1. Oct 30, 2013

### Parmenides

Given:

$$f\left(\frac{az + b}{cz + d}\right) = (cz + d)^kf(z)$$

We can apply:

$$\left( \begin{array}{cc} a & b \\ c & d\\ \end{array} \right) = \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} \right)$$

So that we arrive at the periodicity $f(z+1) = f(z)$. This implies a Fourier expansion:

$$f(z) = \sum_{n=0}^{\infty}c_nq^n$$
Where $q = e^{2{\pi}inz}$

But how to calculate the coefficients $c_n$? Scouring all over the internet, I've seen mention of contour integration and parametrizing along the Half-Plane, but I'm not even sure of the form of the integrand. Ideas would be most appreciated.