  It is generally recognized that the electromagnetic multipolar emission from
magnetars can be used to explain radiation from Soft Gamma Repeaters (SGRs) or
Anomalous X-ray Pulsars (AXPs), but they have little impact on the spindown of
magnetars. We here present a comprehensive analytical solution for the neutron
star multipolar electromagnetic fields and their associated expected
luminosities. We find that for newborn millisecond magnetars, the spin-down
luminosity from higher multipolar components can match or even exceed that from
the dipole component. Such high-intensity radiation will undoubtedly affect
related astrophysical phenomena at the birth of a magnetar. We show that the
spin-down luminosity from multipoles can well explain the majority of Gamma-Ray
Bursts (GRBs) afterglows, from the plateau starting at several hundred seconds
till the normal decay phase lasting for many years. The fitted magnetar
parameters for GRB afterglows are all typical values, with spins in the
millisecond range and magnetic field strengths in the order of $10^{14} -
10^{15}$ Gauss. Our results in turn, provide support for the hypothesis that
GRBs originate from the birth of magnetars with a few millisecond period, thus
deepening our understanding of the complex magnetic field structure and the
equation of state of magnetars.
