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In Fig. 12 we show the fields for m = 1 and m = 2 modes that are created by an initial octupole and excites all odd ℓ modes. Specifically, we show the ℓ = 3 , 5 projections for both m values. We also show the full fields for both m values. Notably, for the case m = 1 the dipole mode is also excited, and eventually dominates the full field at late times. In the m = 2 case the smallest ℓ -mode that is excited is the octupole, such that the full field in the m = 2 case decays more rapidly than in the m = 1 case. We also show the local power indices in Fig. 13. The late time values of the local power indices for each projected ℓ mode approach the same values independently of the value of m , in accordance with Eq. (3). 

In Table IV we re-organize the data appearing in Tables I,II, and III, so that the m -independence of the power-law indices is brought out in a sharper way. Specifically, the late–time decay rates for any allowed excited ℓ ′ mode from an initial ℓ mode is independent of the value of m in all the cases we have examined. 

