  Bernstein-Schwarzman conjectured that the quotient of a complex affine space
by an irreducible complex crystallographic group generated by reflections is a
weighted projective space. The conjecture was proved by Schwarzman and
Tokunaga-Yoshida in dimension 2 for almost all such groups, and for all
crystallographic reflection groups of Coxeter type by Looijenga,
Bernstein-Schwarzman and Kac-Peterson in any dimension. We prove that the
conjecture is true for the crystallographic reflection group in dimension 3 for
which the associated collineation group is Klein's simple group of order 168.
In this case the quotient is the 3-dimensional weighted projective space with
weights 1, 2, 4, 7. The main ingredient in the proof is the computation of the
algebra of invariant theta functions. Unlike the Coxeter case, the invariant
algebra is not free polynomial, and this was the major stumbling block.
