Let $E_4(q)=1+ 240q+ 2160q^2+ 6720q^3+\ldots $ be the Eisenstein series of weight 4, also known as the theta-series of the $E_8$-lattice.
I'm looking for a $\mathbb N$-graded vector space $V$ of graded dimension
$(1, 240, 2160, 6720,\ldots )$.
Moreover, I would like $V$ to be a module for the Virasoro algebra of central charge $c=4$.
I think that this is indeed possible:
Take $1$ times the Verma module of character $1+q+2q^2+3q^3+5q^4+7q^5+\ldots$
plus $239$ times the Verma module of character $q(1+q+2q^2+3q^3+5q^4+\ldots)$
plus $1919$ times the Verma module of character $q^2(1+q+2q^2+3q^3+\ldots)$, etc.
But that's not what I'm looking for.
I'm wondering whether there exists a natural construction that produces the above Virasoro module. Does anybody know of such a construction?
In the above, I said that I wanted central charge $c=4$ and minimal energy $h=0$.
But I'm flexible: if you know a natural construction of a $Vir_c$-module (pick you $c$) whose character is $q^hE_4(q)$, I'll be happy to hear about it.