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Coefficients of Modular Forms

  1. Oct 30, 2013 #1
    Given:

    [tex] f\left(\frac{az + b}{cz + d}\right) = (cz + d)^kf(z)[/tex]

    We can apply:

    [tex]\left( \begin{array}{cc}
    a & b \\
    c & d\\
    \end{array} \right)
    = \left( \begin{array}{cc}
    1 & 1 \\
    0 & 1 \\
    \end{array} \right)[/tex]

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

    [tex]f(z) = \sum_{n=0}^{\infty}c_nq^n[/tex]
    Where [itex]q = e^{2{\pi}inz}[/itex]

    But how to calculate the coefficients [itex]c_n[/itex]? 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.
     
  2. jcsd
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