What are the orbits of a irreducible $m$ dimensional representation of SO(3)

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Chose e.g. $e_1 = (1,0,0,...,0)$. What does the set $\{D^m(g)e_1, g \in SO(3) \}$ look like? ($D^m$ is an m-dimensional irrep of SO(3), $m$ > 3)