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In ths paper we study the VOA $V_{\mathcal{J},r}$ constructed by Ashihara andMiyamoto, We construct simple quotients of$V_{\mathcal{J},r},r\in\mathbb{Z}_{\neq 0}$ explicitly using dual-pair typeconstructions. We also compute the character formula of the simple quotientswhen $r=-2n,n\geq 1$.

Simplicities of VOAs Associated to Jordan Algebras of Type $B$ and Character Formulas for Simple Quotients