  Extreme mass-ratio inspirals (EMRIs) are the most potential sources
detectable by the Laser Interferometer Space Antenna (LISA). To analyze the
influence of higher harmonics on parameter estimation for EMRIs efficiently, we
use the waveform model that the phase trajectories are relativistic flux-based
adiabatic trajectories and the waveforms are constructed by the augmented
analytic kludge method. We perform a Fisher-matrix error analysis of the EMRI
parameters using signals taking into account the motion of the LISA
constellation and higher harmonics of gravitational waves. Our results
demonstrate that including higher harmonics greatly reduces the errors on the
exterior parameters such as inclination angle $\iota$, the luminosity distance
$d_L$, the polarization angle $\psi$, and the initial phase $\Phi_0$, except
for source localization $\Delta \Omega$ when EMRIs face us. However, the
influence of higher harmonics on parameters $(\iota,d_L,\psi,\Phi_0)$ can be
negligible when the inclination angle is above $1.0$. For intrinsic parameters
such as the spin of central black and the masses of binaries, the influence of
higher harmonics can be negligible for any inclination angle. Our findings are
independent of the mass or spin of the EMRI system.
