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This paper describes an analysis of the NuSTAR data of the fastest-rotatingmagnetar 1E 1547$-$5408, acquired in 2016 April for a time lapse of 151 ks. Thesource was detected with a 1-60 keV flux of $1.7 \times 10^{-11}$ ergs s$^{-1}$cm$^{-2}$, and its pulsation at a period of $2.086710(5)$ sec. In 8-25 keV, thepulses were phase-modulated with a period of $T=36.0 \pm 2.3$ ks, and anamplitude of $\sim 0.2$ sec. This reconfirms the Suzaku discovery of the sameeffect at $T=36.0 ^{+4.5}_{-2.5} $ ks, made in the 2009 outburst. These resultsstrengthen the view derived from the Suzaku data, that this magnetar performsfree precession as a result of its axial deformation by $\sim 0.6 \times10^{-4}$, possibly caused by internal toroidal magnetic fields reaching $\sim10^{16}$ G. Like in the Suzaku case, the modulation was not detected inenergies below $\sim 8$ keV. Above 10 keV, the pulse-phase behaviour, includingthe 36 ks modulation parameters, exhibited complex energy dependences: at $\sim22$ keV, the modulation amplitude increased to $\sim 0.5$ sec, and themodulation phase changed by $\sim 65^\circ$ over 10--27 keV, followed by aphase reversal. Although the pulse significance and pulsed fraction wereoriginally very low in $>10$ keV, they both increased noticeably, when thearrival times of individual photons were corrected for these systematicpulse-phase variations. Possible origins of these complex phenomena arediscussed, in terms of several physical processes that are specific toultra-strong magnetic fields.

A NuSTAR confirmation of the 36 ks hard X-ray pulse-phase modulation in the magnetar 1E 1547.0$-$5408