  The brightest cluster radio halo known resides in the Coma cluster of
galaxies. The relativistic electrons producing this diffuse synchrotron
emission should also produce inverse Compton emission that becomes competitive
with thermal emission from the ICM at hard X-ray energies. Thus far, claimed
detections of this emission in Coma are controversial (Fusco-Femiano et al.
2004; Rossetti & Molendi 2004). We present a Suzaku HXD-PIN observation of the
Coma cluster in order to nail down its non-thermal hard X-ray content. The
contribution of thermal emission to the HXD-PIN spectrum is constrained by
simultaneously fitting thermal and non-thermal models to it and a spatially
equivalent spectrum derived from an XMM-Newton mosaic of the Coma field
(Schuecker et al. 2004). We fail to find statistically significant evidence for
non-thermal emission in the spectra, which are better described by only a
single or multi-temperature model for the ICM. Including systematic
uncertainties, we derive a 90% upper limit on the flux of non-thermal emission
of 6.0x10^-12 erg/s/cm^2 (20-80 keV, for photon index of 2.0), which implies a
lower limit on the cluster-averaged magnetic field of B>0.15 microG. Our flux
upper limit is 2.5x lower than the detected non-thermal flux from RXTE
(Rephaeli & Gruber 2002) and BeppoSAX (Fusco-Femiano et al. 2004). However, if
the non-thermal hard X-ray emission in Coma is more spatially extended than the
observed radio halo, the Suzaku HXD-PIN may miss some fraction of the emission.
A detailed investigation indicates that ~50-67% of the emission might go
undetected, which could make our limit consistent with these detections. The
thermal interpretation of the hard Coma spectrum is consistent with recent
analyses of INTEGRAL (Eckert et al. 2007) and Swift (Ajello et al. 2009) data.
