  Indications of disequilibrium throughout the Milky Way (MW) highlight the
need for compact,flexible, non-parametric descriptions of phase--space
distributions of galaxies. We present a new representation of the current Dark
Matter (DM) distribution and potential derived from N-body simulations of the
Milky Way and Large Magellanic Cloud (LMC) system using Basis Function
Expansions (BFEs). We incorporate methods to maximize the physical signal in
the representation. As a result, the simulations of $10^8$ DM particles
representing the MW--LMC system can be described by 354 coefficients. We find
that the LMC induces asymmetric perturbations (odd l, m) to the MW's halo,
which are not well-described by oblate, prolate, or triaxial halos.
Furthermore, the energy in high-order even modes (l,m $\geq$ 2) is similar to
average triaxial halos found in cosmological simulations. As such, the response
of the MW's halo to the LMC must be accounted for in order to recover the
imprints of its assembly history. The LMC causes the outer halo ($\geq$ 30 kpc)
to shift from the disk center of mass (COM) by $\sim$15-25 kpc at present day,
manifesting as a dipole in the BFE and in the radial velocities of halo stars.
The shift depends on the LMC's infall mass, the distortion of the LMC's halo
and the MW halo response. Within 30 kpc, halo tracers are expected to orbit the
COM of the MW's disk, regardless of LMC infall mass. The LMC's halo is also
distorted by MW tides, we discuss the implications for its mass loss and the
subsequent effects on current Magellanic satellites.
