  Axions are a promising dark matter candidate that were motivated to solve the
strong CP problem and that may also address the cosmological matter-antimatter
asymmetry. Axion-photon conversion is possible in the presence of the strong
magnetic fields, and the photon so produced will have energy equal to the axion
mass. Here we report new limits on axionic dark matter obtained from radio
spectra of the Galactic Center magnetar PSR J1745-2900. The magnetar has a
magnetic field of $1.6\times10^{14}$ G that interacts with a dark matter
density $2\times10^5$ to $2\times10^9$ times greater than the local dark matter
encountered by terrestrial haloscopes, depending on the Galactic dark matter
profile. No significant spectral features are detected across 62% of the axion
mass range 4.1-165.6 $\mu$eV (1-40 GHz). The interpretation of flux limits into
limits on the two-photon coupling strength $g_{a\gamma\gamma}$ depends on the
magnetospheric conversion model and on the dark matter density at the Galactic
Center. For a standard dark matter profile, we exclude axion models with
$g_{a\gamma\gamma}> $ 6-34 $\times 10^{-12}$ GeV$^{-1}$ with 95% confidence
over the mass ranges 4.2-8.4, 8.9-10.0, 12.3-16.4, 18.6-26.9, 33.0-62.1,
70.1-74.3, 78.1-80.7, 105.5-109.6, 111.6-115.2, 126.0-159.3, and 162.5-165.6
$\mu$eV. For the maximal dark matter cusp allowed by stellar orbits near Sgr
A*, these limits reduce to $g_{a\gamma\gamma} > $ 6-34 $ \times10^{-14}$
GeV$^{-1}$, which exclude some theoretical models for masses $> 33$ $\mu$eV.
Limits may be improved by modeling stimulated axion conversion, by ray-tracing
conversion pathways in the magnetar magnetosphere, and by obtaining deeper
broad-band observations of the magnetar.
