  The quadratic Jahn-Teller effect of C$_{60}^{n-}$ ($n=$ 1-5) is investigated
from the first principles. Employing the density functional theory calculations
with hybrid functional, the quadratic vibronic coupling constants of C$_{60}^-$
were derived. The warping of the adiabatic potential energy surface of
C$_{60}^-$ by the quadratic vibronic coupling is estimated about 2 meV, which
is much smaller than the Jahn-Teller stabilization energy ($\approx$ 50 meV).
Because of the selection rule and the vibronic reduction, the quadratic
coupling slightly modifies the vibronic states of C$_{60}$ anions.
Particularly, in the case of C$_{60}^{3-}$, parity and symmetry selection rule
significantly reduces the effect of quadratic coupling on vibronic states. The
present results confirm that the low-energy vibronic dynamics of C$_{60}^{n-}$
is of pseudorotational type.
