  In the standard basis exact expressions for the components of SIC vectors
(belonging to a symmetric informationally complete POVM) are typically very
complicated. We show that a simple transformation to a basis adapted to the
symmetries of a fiducial SIC vector can result in a massive reduction in
complexity. We rely on a conjectural number theoretic connection between SICs
in dimension $d_j$ and SICs in dimension $d_{j+1} = d_j(d_j-2)$. We focus on
the sequence 5, 15, 195, ... . We rewrite Zauner's exact solution for the SIC
in dimension 5 to make its simplicity manifest, and use our adapted basis to
convert numerical solutions in dimensions 15 and 195 to exact solutions.
Comparing to the known exact solutions in dimension 15 we find that the
simplification achieved is dramatic. The proof that the exact vectors are
indeed SIC fiducial vectors, also in dimension 195, is a long calculation
guided by the standard ray class hypothesis about the algebraic number fields
generated by the SICs. We conjecture that our result generalizes to every
dimension in the particular sequence we consider.
