Define $A(0), A(1), A(2) \dots$ in ${\bf Z}/3[[x]]$ as follows. For $n$ in $\bf N$ let $s=3^{2n+1}$. Then $A(n) = \sum a_kx^k$ where $a_k$ is the mod 3 reduction of the number of representations of $k$ by the principal positive binary quadratic form of discriminant $-s$, and the sum runs over all $k$ prime to 3.

Example 1__ When $n=0$, we take the form to be $u^2+uv+v^2$. The number of representations of any non-zero $k$ by this form is a multiple of 6, and so $A(0)=0$.

Example 2__ $-A(1)$ is the mod 3 reduction of (the expansion of) the usual weight 12 cusp form for the full modular group.

Experiment suggests a recursion for the $A(n)$ which would allow one to write each $A(n)$, $n>0$, as a polynomial of degree $(s-3)/24$ in $A=A(1)$. Explicitly I ask if the following holds:

(*)___ $A(n+2)=(A^{3s}+A^{2s}+1)A(n+1)-A^{2s}A(n)$

If (*) holds, then for example:

$A(2)=A^{10}+A^7+A$

$A(3)-A(2) = \sum A^k$, $k$ in $\{91,88,82,64,61\}$

$A(4)-A(3) = \sum A^k$, $k$ in $\{820,817,811,793,790,739,736,730,577,574,568,550,547\}$

I've experimentally verified the above 3 identities, and I'm sure that modular forms could be used to settle the truth of (*) for any fixed $n$.

EDIT: As far as the weaker question of finding a proof that each $A(n)$ is a polynomial in $A$, there is a possibly relevant article by Skoruppa. (arXiv:0807.4694v2---Reduction mod $\ell$ of theta series of level $\ell^n$). But his main result is restricted to characteristic >3. I quote the result:

It is proved that the theta series of an even lattice whose level is a power of a prime $\ell$ is congruent modulo $\ell$ to an elliptic modular form of level one.

(Note that every elliptic modular form over $\bf Q$ of level 1 has as its modulo 3 reduction a polynomial in $A$. So one would want to extend Skoruppa's result to $\ell=3$).