Fourier series analogue for Modular Group?

is there a fourier series analogue to the modular group ?=?

$$ zmapstofrac{az+b}{cz+d},$$

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) $