  We consider the operator algebra $\mathscr A$ on $\mathscr S(\mathbb R^n)$
generated by the Shubin type pseudodifferential operators, the Heisenberg-Weyl
operators and the lifts of the unitary operators on $\mathbb C^n$ to
metaplectic operators.
  With the help of an auxiliary operator in the Shubin calculus, we find trace
expansions for these operators in the spirit of Grubb and Seeley. Moreover, we
can define a noncommutative residue generalizing that for the Shubin
pseudodifferential operators and obtain a class of localized equivariant traces
on the algebra.
