The eigenvalues of \(\Delta_{\mathbb{S}^d_R}\) are \(-\frac{l(l+d-1)}{R^2}\) where \(l\in \mathbb{N}_0\). The corresponding eigenspaces are the spherical harmonics of degree \(l\).
The dimension of the \(l\)-th eigenspace is given by
\[ n_l = \Bigl(1+\frac{2l}{d-1}\Bigr)\binom{l+d-2}{l}. \]