Unfortunately, the current implementation of spherical harmonics in Sage suffers from various issues. One is the sign convention, which is different from the standard one, as you pointed out. Another issue is getting false results for some parameters, like
sage: theta, phi = var('theta phi') sage: spherical_harmonic(1, 1, theta, phi) -1/4*sqrt(3)*sqrt(2)*sqrt(-sin(theta)^2)*e^(I*phi)/sqrt(pi) The term sqrt(-sin(theta)^2) in the above expression is plain false. It should be sin(theta). This issue is related to the bug reported in https://trac.sagemath.org/ticket/25034 As a workaround, I would recommend to use the function spin_weighted_spherical_harmonic with the spin weight set to zero instead of spherical_harmonic. The function spin_weighted_spherical_harmonic is not part of the main SageMath, but of the SageMath package kerrgeodesics_gw <https://pypi.org/project/kerrgeodesic-gw/>. So you have to install the later via sage -pip install kerrgeodesic_gw NB: on CoCalc, you have to add the option --user: sage -pip install --user kerrgeodesic_gw Then, you will get (the first argument, i.e. 0, is the spin weight; if zero, one gets the standard spherical harmonics): sage: from kerrgeodesic_gw import spin_weighted_spherical_harmonic sage: spin_weighted_spherical_harmonic(0, 2, 1, theta, phi) -1/4*sqrt(15)*sqrt(2)*cos(theta)*e^(I*phi)*sin(theta)/sqrt(pi) sage: spin_weighted_spherical_harmonic(0, 1, 1, theta, phi) -1/4*sqrt(6)*e^(I*phi)*sin(theta)/sqrt(pi) Both results agree with the standard values, as given e.g. in https://en.wikipedia.org/wiki/Table_of_spherical_harmonics Eric. -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at https://groups.google.com/group/sage-support. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-support/633d5f7d-dbb1-441d-9688-cd12a7ac7720%40googlegroups.com. For more options, visit https://groups.google.com/d/optout.