7 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 −3 3 3.05 3.1 3.15 3.2 log$_{10}$ M / t 0 n 0 0 0.5 1 1.5 2 2.5 3 3.5 x 10 −3 5 5.05 5.1 5.15 5.2 log$_{10}$ M / t 0 n 2 0 0.5 1 1.5 2 2.5 3 −18 −16 −14 −12 −10 −8 −6 −4 log$_{10}$ t / M log 10 |  0  $_{ψ}$ 4  | 100Kx100 80Kx80 64Kx64 2.6 2.65 2.7 2.75 2.8 2.85 2.9 −16 −15 −14 t −7 3 3.05 3.1 3.15 −17.2 −17 −16.8 −16.6 Initial Projected Projected Projected Projected Projected Projected ℓ ′ mode ℓ = 0 ℓ = 1 ℓ = 2 ℓ = 3 ℓ = 4 ℓ = 5 0 3.007 * 5.01 * 7 * 1 * 5.002 * 7.003 * 9 2 3.005 * 7.009 * 9.01 * 3 * 5.002 * 9.009 * 11.008 4 5.01 * 7.002 * 11.008 * FIG. 4: The local power indices 0 n 0 (upper panel) and 0 n 2 (lower panel) as functions of the time t . FIG. 5: The ℓ = 4 multipole projections, for initial data of a pure monopole field as functions of time. Shown are the fields with three different grid densities, 64 K × 64 (dash–dotted curve), 80 K × 80 (dashed curve), and 100 K × 100 (solid curve). The inserts show the same for segments of the full data, in addition to a reference curve ∼ t − $^{7}$. TABLE I: Scattering matrix for azimuthal modes. Late–time power law indices ℓ ′ n$_{ℓ}$ for the ℓ –projections of the fields for pure azimuthal ( m = 0) mode initial data sets, ℓ ′ = 0 , 1 , 2 , 3 , 4. This table includes both “down” and “up” excitations. Asterisks relate to kinematically disallowed states. All figures are significant.