  The effects higher order harmonics have been self-consistently included in
the derivation of the electron branch of the electron Geodesic Acoustic Mode
(el-GAM) in an Electron-Temperature-Gradient (ETG) turbulence background. The
work is based on a two-fluid model including finite $\beta$-effects while
retaining non-adiabatic ions. In solving the linear dispersion relation, it is
found that the due to the coupling to the $m=2$ mode the real frequency may be
significantly altered and yield higher values.
