Calculating Coefficients of Modular Forms

[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.

 

[kurt.physics]

If we have f(z) = \sum_{n=0}^{\infty}c_n q^n with q = e^{2{\pi}inz}, then we can use the definition of a Fourier series to calculate the coefficients c_n. We have that c_n = \frac{1}{2{\pi}}\int_0^{2{\pi}}f(z)q^{-n}dz where q^{-n} = e^{-2{\pi}inz}. This integral can be solved using contour integration techniques. A possible parametrization is along the half-plane, and the integrand has the form f(z)q^{-n}.