In the above, N ′ l is the number of positive roots of the Legendre polynomial, M ′ t = M t /2, and w 0 is the Gauss-Legendre quadrature weight when s n = 0. We used the symmetry of {s n } about s = 0 and the antipodal symmetry of the spherical t-design where every tm with z-component greater than or equal to 0 (upper hemisphere) has a corresponding vector -t m (lower hemisphere) . The symmetry cancels imaginary terms involving sine functions.