is a compact, rapidly rotating bar. The majority of unstable modes belong to a discrete spiral family that bifurcates from a van Kampen mode with $\omega \approx
0.43$. I have labeled these modes by S1,$\cdots$,S6. The number of density peaks along the spiral arms is proportional to the integer number in the mode name. Both B2 and G are double peaked spirals but I have classified B2 as a bar mode, and collected it with B1 in a two member family, for it takes a bar-like structure when it is stabilized by decreasing $\alpha$. I classify mode G as an isolated mode because it does not behave similar to either of S- or B-modes as the model parameters vary. There is another isolated mode in the spectrum, C1, which exhibits a spiral pattern. By decreasing $\lambda$, mode C1 joins a new family of spiral modes, which are accumulated near the galactic center (see §\[sec:variations-of-lambda\]).

Reducing $\alpha$ increases the abundance of dark matter and according to [@T81] and JH the growth rate of modes should decrease. My calculations show that by reducing $\alpha$, spiral modes are affected sooner and more effective than the bar mode, and they join to the stationary 