  We report the detection of the lens cluster of the gravitational lens (GL)
system Q0957+561 from a deep observation with the Advanced CCD Imaging
Spectrometer on-board the Chandra X-ray Observatory. Intracluster X-ray
emission is found to be centered 4.3 +/- 1.3 arcsec east and 3.5(-0.6,+1.3)
arcsec north of image B, nearer than previous estimates. Its spectrum can be
modeled well with a thermal plasma model consistent with the emission
originating from a cluster at a redshift of 0.36. Our best-fit estimates of the
cluster temperature of T_e = 2.09(-0.54,+0.83) keV (90 percent confidence) and
mass distribution of the cluster are used to derive the convergence parameter
kappa, the ratio of the cluster surface mass density to the critical density
required for lensing. We estimate the convergence parameter at the location of
the lensed images A and B to be kappa_A = 0.22(+0.14,-0.07) and kappa_B =
0.21(+0.12,-0.07), respectively (90 percent confidence levels). The observed
cluster center, mass distribution and convergence parameter kappa provide
additional constraints to lens models of this system. Our new results break a
mass-sheet degeneracy in GL models of this system and provide better
constraints of ~ 29 percent (90 percent confidence levels) on the Hubble
constant. We also present results from the detection of the most distant X-ray
jet (z = 1.41) detected to date. The jet extends approximately 8 arcsec NE of
image A and three knots are resolved along the X-ray jet with flux densities
decreasing with distance from the core. The observed radio and optical flux
densities of the knots are fitted well with a synchrotron model and the X-ray
emission is modeled well with inverse Compton scattering of Cosmic Microwave
Background photons by synchrotron-emitting electrons in the jet.
