Panel a: growth rates ReΓ and ( panel b) oscillation frequencies ImΓ of the symmetric (solid) and anti-symmetric (dashed) fastest-growing dynamo eigenmodes in a spherical halo as a function of R_αh, with all the remaining parameters fixed at their fiducial values shown in Table 1. Panels c and d: the magnitudes of the expansion coefficients a_i in Eq. (59) for the dipolar and quadrupolar modes, respectively. We note that each sharp bend in ReΓ that occurs as R_αh changes is connected with an intersection of two curves a_i with distinct values of i.