  Let $V$ be a finite-dimensional representation of the complex circle
$\mathbb{C}^\times$ determined by a weight vector $\mathbf{a}\in\mathbb{Z}^n$.
We study the Hilbert series $\operatorname{Hilb}_{\mathbf{a}}(t)$ of the graded
algebra $\mathbb{C}[V]^{\mathbb{C}_{\mathbf{a}}^\times}$ of polynomial
$\mathbb{C}^\times$-invariants in terms of the weight vector $\mathbf{a}$ of
the $\mathbb{C}^\times$-action. In particular, we give explicit formulas for
$\operatorname{Hilb}_{\mathbf{a}}(t)$ as well as the first four coefficients of
the Laurent expansion of $\operatorname{Hilb}_{\mathbf{a}}(t)$ at $t=1$. The
naive formulas for these coefficients have removable singularities when weights
pairwise coincide. Identifying these cancelations, the Laurent coefficients are
expressed using partial Schur polynomial that are independently symmetric in
two sets of variables. We similarly give an explicit formula for the
$a$-invariant of $\mathbb{C}[V]^{\mathbb{C}_{\mathbf{a}}^\times}$ in the case
that this algebra is Gorenstein. As an application, we give methods to identify
weight vectors with Gorenstein and non-Gorenstein invariant algebras.
