 For states of higher angular momentum quantum number formula_3 and a higher dimension of formula_1 the number of possible rotations or basis transformations in Hilbert space grows and so does the number of possible orthogonal representations that can be constructed on the basis of the formula_1-dimensional spherical harmonics basis set.There is more freedom to choose a representation that fits the point group symmetry of the problem.The cubic representations that are listed in the table are a result of the transformations, which are 45° 2D rotations and a 90° rotation to the real axis if necessary, like

A substantial number of the spherical harmonics are listed in the Table of spherical harmonics.