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The general linear group $GL_n(\\mathbb{F}_q)$ acts on $\\Omega$; write $SI \\subseteq \\Omega$ for the ideal generated by $GL_n(\\mathbb{F}_q)$-invariants with vanishing constant term. The {\\em $GL_n(\\mathbb{F}_q)$-superspace coinvariant ring} is the quotient $SR := \\Omega/SI$. We calculate the bigraded Hilbert series of $SR$ and give an operator-theoretic characterization of the inverse system $SI^\\perp$. 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