  Neutrinos emitted from a supernova may undergo flavor conversions almost
immediately above the core, with possible consequences for supernova dynamics
and nucleosynthesis. However, the precise conditions for such fast conversions
can be difficult to compute and require knowledge of the full angular
distribution of the flavor-dependent neutrino fluxes, that is not available in
typical supernova simulations. In this paper, we show that the overall flavor
evolution is qualitatively similar to the growth of a so-called `zero mode',
determined by the background matter and neutrino densities, which can be
reliably predicted using only the second angular moments of the electron lepton
number distribution, i.e., the difference in the angular distributions of
$\nu_e$ and $\bar{\nu}_e$ fluxes. We propose that this zero mode, which neither
requires computing the full Green's function nor a detailed knowledge of the
angular distributions, may be useful for a preliminary diagnosis of possible
fast flavor conversions in supernova simulations with modestly resolved angular
distributions
