TITLE: I'm looking for a Virasoro-module whose character is  1+ 240q+ 2160q^2+ 6720q^3...
QUESTION [13 upvotes]: 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?

Variant:
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.

REPLY [5 votes]: Would you be happy with a supercharacter $str \ q^{L(0)-c/24}$ as opposed to the usual graded dimension ${\rm tr} \ q^{L(0)-c/24}$? Also, from the $q$-exapansion of $E_4$, it seems to me that the correct central charge should be zero, not $4$. 
You could  take $V(E_8)$, the $E_8$ 
 lattice vertex algebra  of central charge $c=8$, and tensor it with the $\mathbb{Z}_2$-graded  symplectic fermion  vertex 
superalgebra $\mathcal{SF}^8$ of central charge $c'=-8$. The tensor product $V(E_8) \otimes \mathcal{SF}^8$ is $\mathbb{Z}_2$-graded by the fermionic part. 
Since ${\rm tr}({\mathcal{SF}^8}) q^{L(0)-c'/24}=q^{1/3} \prod_{i=1}^\infty (1+q^i)^8$,
after taking the supertrace of the tensor product, you will clear the denominator in 
$\frac{\theta_{E_8}}{\eta^8}$, and end up with the supertrace equal $\theta_{E_8}$. Actually you can think of it as a virtual character (the Virasoro generator is even).
This construction is perhaps silly, but it has some motivation. In rational CFT, all characters are of weight zero. In rational SCFT  there are also supercharacters, again of weight zero.  Symplectic fermions $\mathcal{SF}^{2d}$, $d \geq 1$ are not rational, but "quasi-rational" or sometimes called "logarithmic", and their supercharacters exhibit "higher weight" phenomena.