  We consider a trapped Bose--Einstein condensate (BEC) with a highly quantized
vortex. For the BEC with a doubly, triply or quadruply quantized vortex, the
numerical calculations have shown that the Bogoliubov--de Gennes equations,
which describe the fluctuation of the condensate, have complex eigenvalues. In
this paper, we obtain the analytic expression of the condition for the
existence of complex modes, using the method developed by Rossignoli and
Kowalski [R. Rossignoli and A. M. Kowalski, Phys. Rev. A 72, 032101 (2005)] for
the small coupling constant. To derive it, we make the two-mode approximation.
With the derived analytic formula, we can identify the quantum number of the
complex modes for each winding number of the vortex. Our result is consistent
with those obtained by the numerical calculation in the case that the winding
number is two, three or four. We prove that the complex modes always exist when
the condensate has a highly quantized vortex.
