  The possibility that our space is multi - rather than singly - connected has
gained a renewed interest after the discovery of the low power for the first
multipoles of the CMB by WMAP. To test the possibility that our space is a
multi-connected spherical space, it is necessary to know the eigenmodes of such
spaces. Excepted for lens and prism space, and in some extent for dodecahedral
space, this remains an open problem. Here we derive the eigenmodes of all
spherical spaces. For dodecahedral space, the demonstration is much shorter,
and the calculation method much simpler than before. We also apply to
tetrahedric, octahedric and icosahedric spaces. This completes the knowledge of
eigenmodes for spherical spaces, and opens the door to new observational tests
of cosmic topology.
  The vector space V^k of the eigenfunctions of the Laplacian on the
three-sphere S^3, corresponding to the same eigenvalue \lambda_k = -k (k+2),
has dimension (k+1)^2. We show that the Wigner functions provide a basis for
such space. Using the properties of the latter, we express the behavior of a
general function of V^k under an arbitrary rotation G of SO(4). This offers the
possibility to select those functions of V^k which remain invariant under G.
  Specifying G to be a generator of the holonomy group of a spherical space X,
we give the expression of the vector space V_X^k of the eigenfunctions of X. We
provide a method to calculate the eigenmodes up to arbitrary order. As an
illustration, we give the first modes for the spherical spaces mentioned.
