modes B1, B2, G, S1, S2 and S3. It is seen that the patterns of S-modes rotate slower and become more extensive as the mode number increases. Mode S6, which is at the bifurcation point of the spiral family, has the largest extent. Eight wave packets of this mode are distributed by a phase shift of 90 degrees along major spiral arms. Mode G has at most two density peaks on its major spiral arms but the magnitude of its second peak increases as the disk is cooled. Modes B1, B2, S1, S2 and S3 are, respectively, analogous to modes A, B, C, E and F of a Gaussian disk explored in @T81. There are three low-speed modes in Figure 11 of @T81 that have not been labeled, but they are analogs of modes S4, S5 and S6. Mode G and Toomre’s mode D also have some similarities but are of different origins (see §\[sec:final-discussions\]). None of them can be stabilized only by increasing the fraction of dark matter.

Figure \[pic:spectrum-m2-unstable\][*a*]{} shows that the fundamental mode obtained by JH coincides with mode B1. The wave pattern of mode B1 (displayed in Figure \[pic:mode-shape-m2\]) is identical to the mode shape computed 