  Black holes carry more information about the microstates than just the total
degeneracy. As a concrete example, the Z(N)-twined helicity trace indices for
1/4-BPS black holes of the CHL models allow extracting information about the
distribution of the Z(N) charges among the black hole microstates. The number
of black hole microstates carrying a definite eigenvalue under the generator of
the Z(N) twining group must be positive. This leads to a specific prediction
for the signs of certain linear combinations of Fourier coefficients of Siegel
modular forms. We explicitly test these predictions for low charges. In the
D1-D5-P duality frame, we compute the appropriate hair removed partition
functions and show the positivity of the appropriate Fourier coefficients for
low charges. We present various consistency checks on our computations.
