Spherical Harmonics

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Hello,

I was wondering the difference between Sage's spherical_harmonic (as well as Maxima's spherical_harmonic) and Mathematica's SphericalHarmonicY for the same arguments.

Sage's

print(spherical_harmonic(1,1,pi/3,pi/6).n())
print(spherical_harmonic(1,-1,pi/3,pi/6).n())

gives the following result:

0.149603355150537 - 0.259120612103502*I
-0.259120612103502 + 0.149603355150537*I

Maxima's

print(maxima.spherical_harmonic(1,1,pi/3,pi/6).n())
print(maxima.spherical_harmonic(1,-1,pi/3,pi/6).n())

gives

0.259120612103502 + 0.149603355150537*I
-0.259120612103502 + 0.149603355150537*I

and Mathematica,

SphericalHarmonicY[1, 1, Pi/3, Pi/6] // N
SphericalHarmonicY[1, -1, Pi/3, Pi/6] // N

gives

-0.259121 - 0.149603*I
0.259121 - 0.149603*I

I have come across the convention difference between Maxima and Mathematica in some textbooks-- that is the (-1)^m factor, but I am not quite sure if I get it for between Sage's and Mathematica's.

So, what is the mathematical relationship between Sage's spherical_harmonic and Mathematica's SphericalHarmonicY in terms of l and m?

Thanks alot.