Let $E\_4(q)=1+ 240q+ 2160q^2+ 6720q^3+\ldots $ be the [Eisenstein series](http://en.wikipedia.org/wiki/Eisenstein_series#Products_of_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?




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**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.



I'm looking for a Virasoro-module whose character is  1+ 240q+ 2160q^2+ 6720q^3...
What I was asking for cannot be realized... for a simple and stupid reason:


The coefficients of the Verma module of the Virasoro algebra are given by the partition function $p(n)=\{1,1,2,3,5,7,11,...\}$.
Now, the growth of the partition function is given by
$$p(n) \sim \frac {1} {4n\sqrt{3}} e^{\pi \sqrt {\frac{2n}{3}}} \mbox { as } n\rightarrow \infty.$$
(see e.g.  [wikipedia](http://en.wikipedia.org/wiki/Partition_(number_theory)#Partition_function_formulas)) and is much faster than the growth of the coefficients of the theta function of the $E\_8$-lattice (which is polynomial).


So my prescription "take one times the Verma module [...] plus 239 times the Verma module [...] plus 1919 times ..." will run into negative numbers at some point.


