  We study the four-gluon scattering amplitude in the high energy limit of QCD
written in terms of its conformal expansion. We highlight the need to include
both even and odd conformal spin contributions in order to map it to an
iterative representation in rapidity and transverse momentum space which we
have evaluated numerically. By Fourier expanding in a set of three azimuthal
angles, we find a new form for the amplitude in terms of $_4F_3$ hypergeometric
functions. An alternative formulation is possible when connecting this Fourier
expansion with Bessel kernels studied in analytic number theory.
