We report on a 350-ks NuSTAR observation of the magnetar 1E 1841-045 taken in 2013 September. During the observation, NuSTAR detected six bursts of short duration, with $T_{90}<1$ s. An elevated level of emission tail is detected after the brightest burst, persisting for $sim$1 ks. The emission showed a power-law decay with a temporal index of 0.5 before returning to the persistent emission level. The long observation also provided detailed phase-resolved spectra of the persistent X-ray emission of the source. By comparing the persistent spectrum with that previously reported, we find that the source hard-band emission has been stable over approximately 10 years. The persistent hard X-ray emission is well fitted by a coronal outflow model, where $e^{+/-}$ pairs in the magnetosphere upscatter thermal X-rays. Our fit of phase-resolved spectra allowed us to estimate the angle between the rotational and magnetic dipole axes of the magnetar, $alpha_{mag}=0.25$, the twisted magnetic flux, $2.5times10^{26}rm G cm^2$, and the power released in the twisted magnetosphere, $L_j=6times10^{36}rm erg s^{-1}$. Assuming this model for the hard X-ray spectrum, the soft X-ray component is well fit by a two-blackbody model, with the hotter blackbody consistent with the footprint of the twisted magnetic field lines on the star. We also report on the 3-year Swift monitoring observations obtained since 2011 July. The soft X-ray spectrum remained stable during this period, and the timing behavior was noisy, with large timing residuals.
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