is there a fourier series analogue to the modular group ?=?
$$ z\mapsto\frac{az+b}{cz+d},$$
for example we could expand any function that satisfies the modular equation
$$ f(\frac{az+b}{cz+d}) =f(z) $$
in terms of simpler function which are modular , just in the same case as we expand any periodic function
$$ f(x+T)=f(x) $$
in terms of exponential funtions (Fourier series) $ exp(inx) $