  A prominent class of model FQH ground states is those realized as correlation
function of $\mathbb{Z}_k^{(r)}$-algebras. In this paper, we study the
interplay between these algebras and their corresponding wavefunctions. In the
hopes of realizing these wavefunctions as a unique densest zero energy state,
we propose a generalization for the projection Hamiltonians
$\mathscr{H}_{k+1}^{2r-1}$. Finally, using techniques from invariants of binary
forms, an ansatz for computation of correlations $\langle
\psi(z_1)\cdots\psi(z_{2k})\rangle \prod_{i<j}(z_i-z_j)^{2r/k}$ is devised. We
provide some evidence that, at least when $r=2$, our proposed Hamiltonian
realizes $\mathbb{Z}_k^{(2)}$-wavefunctions as a \emph{unique} ground state.
