  The spin evolution of a Bose-Einstein condensate starting from a mixture of
two or three groups of $^{52}$Cr (spin-3) atoms in an optical trap has been
studied theoretically. The initial state is so chosen that the system does not
distinguish up and down. In this choice, the deviation caused by the
single-mode approximation is reduced. Moreover, since the particle number is
given very small (N=20), the deviation caused by the neglect of the long-range
dipole force is also reduced. Making use of these two simplifications, a
theoretical calculation beyond the mean field theory is performed. The
numerical results are help to evaluate the unknown strength $g_0$.
