Let D in Z/3[[x]] be sum ((a_n)(x^n)) where the sum runs over all n prime to 6 and a_n is the mod 3 reduction of the number of ideals of norm n in the ring of integers of Q(root(-3)). (So D=x+2(x^7)+2(x^13)+2(x^19)+x^25+(higher degree terms).)
There are formal Hecke operators T_p: Z/3[[x]]-->Z/3[[x]] for all primes p other than 3. Clearly the T_p with p=2 mod 3 annihilate D. Experimentally I find that they annihilate D^7, D^61 and D^547 as well.
Question: If k=(1+3^m)/4 with m odd, do the T_p with p=2 mod 3 annihilate D^k?
Remark: D is the reduction of the expansion of the modular form (eta(2z))^12, but it's not clear that this helps.
EDIT: If I've made no mistake, Noam's formula can be tweaked to show that D^k=V-U where U and V are the mod 3 reductions of the theta series attached to the binary quadratic forms 36xx+6xy+kyy and 36xx+30yy+(k+6)yy. This raises the question of which linear combos of theta series of primitive binary quadratic forms lie in the space spanned by the powers of D. I think similar questions have been answered by Nicolas and Serre (ell=2, level=1) and perhaps by Bellaiche and or Serre (ell=3, level=1).