  In this paper, we consider the decomposition of theta series for lattice
cosets of ternary lattices. We show that the natural decomposition into an
Eisenstein series, a unary theta function, and a cuspidal form which is
orthogonal to unary theta functions correspond to the theta series for the
genus, the deficiency of the theta series for the spinor genus from that of the
genus, and the deficiency of the theta series for the class from that of the
spinor genus, respectively. These three pieces are hence invariants of the
genus, spinor genus, and class, respectively, extending known results for
lattices and verifying a conjecture of the first author and Haensch. We
furthermore extend the definition of $p$-neighbors to include lattice cosets
and construct an algorithm to compute respresentatives for the classes in the
genus or spinor genus via the $p$-neighborhoods.
