Modular forms can also be viewed as functions in the 2d projective space with a special symmetry:

let f(ax, ay) = a^k f(x,y) be a homogeneous function in the plane x,y for a scalar a, then f can also be written as:

f(x,y) = f(y*x/y, y) = y^k f(x/y,1)

and the symmetry by a matrix transformation in the plane

x-> ax + by and y-> cx+dy then transforms the function f as:

f(ax+by, cx+dy) = (cx+dy)^k f((ax+by)/(cx+dy),1)

now introduce z=x/y and f(x/y,1)=F(z) then

F((az+b)/(cz+d)) = (cz+d)^k F(z)

Projective spaces are cool ;-)

just my 2 ct