  We report on the results of a $Suzaku$ observation of the plasma in the
filament located between the two massive clusters of galaxies Abell 399 and
Abell 401. Abell 399 ($z$=0.0724) and Abell 401 ($z$=0.0737) are expected to be
in the initial phase of a cluster merger. In the region between the two
clusters, we find a clear enhancement in the temperature of the filament plasma
from 4 keV (expected value from a typical cluster temperature profile) to
$kT\sim$6.5 keV. Our analysis also shows that filament plasma is present out to
a radial distance of 15' (1.3 Mpc) from a line connecting the two clusters. The
temperature profile is characterized by an almost flat radial shape with
$kT\sim$6-7 keV within 10' or $\sim$0.8 Mpc. Across $r$=8'~from the axis, the
temperature of the filament plasma shows a drop from 6.3 keV to 5.1 keV,
indicating the presence of a shock front. The Mach number based on the
temperature drop is estimated to be ${\cal M}\sim$1.3. We also successfully
determined the abundance profile up to 15' (1.3 Mpc), showing an almost
constant value ($Z$=0.3 solar) at the cluster outskirt. We estimated the
Compton $y$-parameter to be $\sim$14.5$\pm1.3\times10^{-6}$, which is in
agreement with $Planck$'s results (14-17$\times10^{-6}$ on the filament). The
line of sight depth of the filament is $l\sim$1.1 Mpc, indicating that the
geometry of filament is likely a pancake shape rather than cylindrical. The
total mass of the filamentary structure is $\sim$7.7$\times10^{13}~\rm
M_{\odot}$. We discuss a possible interpretation of the drop of X-ray emission
at the rim of the filament, which was pushed out by the merging activity and
formed by the accretion flow induced by the gravitational force of the
filament.
