}$ for the diagonalization is limited to the lowest Landau levels (LLL),  i.e., all the $\phi _{m_{i}k_{i}}$ contained in $\ \psi _{\alpha }$ have $k_{i}=0$. However, step by step, BFs of higher Landau levels will mixed in. In order to speed up the convergency, the $\Omega _{0}$ in eq.(2) is considered as a variational parameter to optimize the lower eigenenergies emerged from the diagonalization.

\[w\]          In the following calculation, we have m\*=0.067$m_{e}$, $\hbar \omega _{0}$=3meV, $\varepsilon _{r}$=12.4 (for a GaAs dot).  To show the convergency, as an example, the lowest eigenenergies with $L$=82 are obtained as $436.895,$ $436.806$ and $436.760$meV when the number of BFs are 6000, 9000, and 12000, respectively. One can see that the convergency is not very good. However, the densities calculated below by using 6000, 9000 and 12000 BFs are indistinguishable (e.g. in Fig.1). Since we are mainly interested in the qualitative aspect, the accuracy that we have achieved is sufficient.

         After the diagonalization the eigenstates are obtained. The series of states  having the same $L$ is labeled as ($L$)$%
_{i} $. The $i=1$ state (the lowest of the $L$-series) is called a first-state.

The eigenwavefunctions of a 13-electron system are complicated. In order to extract 