Convergence of the numerically computed eigenvalues E_nl  =  E_0, 0 (top) and E_nl  =  E_40, 6 (bottom) for three different discretization schemes. We find naive-FD, effective in the halo case, to show strongly degraded convergence properties under cylindrical symmetry. While the ad hoc corrected finite difference scheme (see text) recovers quadratic convergence, it is outperformed by the exponential convergence of the parity aware Chebyshev pseudospectral approach cheb-parity, which is the method of choice for this work. Insets: High-resolution (N = 4096) reference modes R_nl, the eigenvalue of which is used as E_ref.