Due to the multipleness of this problem, I am offering more points than a normal Physics question. I can repost question again for additional points.
Starting from the definition of the spherical harmonics in terms of Legendre polynomials, derive Y30(θ,φ), Y31(θ,φ), and Y32(θ,φ). Write down the differential operators corresponding to the L2 and Lz operators in spherical coordinates. By explicitly acting these operators on the spherical Harmonics check that spherical Harmonics are indeed eigen states of the L2 and Lz operators. What are their eigen values? Are these values consistent with what you expect from the bra-ket analysis?