Let f in Z/3[[x]] be the mod 3 reduction of the Fourier expansion of the normalized weight 12 cusp form delta for the full modular group. The exponents appearing in f are all 1 mod 3. Fix k>0 and prime to 3. If j is 1 or 2, let S(j) consist of all primes p for which the coefficient of x^p in f^k is j. It's well known that if k=1, S(1) is empty and S(2) consists of all p that are 1 mod 3.
I find experimentally that similar results hold when k is 2,4,5 or 10. Indeed it seems:
When k=2, S(1) is all p that are 2 mod 9, S(2) all p that are 5 mod 9
When k=4, S(1) is all p that are 4 or 7 mod 9, S(2) is empty
When k=5, S(1) is all p that are 5,7 or 20 mod 27, S(2) all p that are 8,11, or 23 mod 27
When k=10, S(1) is all p that are 13 or 25 mod 27, S(2) all p that are 16 or 22 mod 27.
I have little doubt that these results hold. But are they known, and is there a reference?
EDIT: For the case of reduction mod 2 rather than mod 3, see my recent question, "Does this theorem of Hasse....?" . But it seems less likely that techniques from the theory of binary and ternary quadratic forms can yield a proof of the above "results".
FURTHER EDIT: Here's a sketch of a proof of the first result. The space of Fourier series of weight 2 cusp forms for gamma_0 (9) has a basis of Eisenstein elements F,G, and H lying in Z[[x^3]], xZ[[x^3]], and x^2 Z[[x^3]] respectively. In Z[[x]], F is congruent to 1 mod 12 x^3. Furthermore the coefficient of x^n in G is sigma_1(n) when n is 1 mod 3, while the coefficient of x^n in H is (1/3)(sigma_1 (n)) when n is 2 mod 3.
Let C=x-8x^4+20x^7+.. be the Fourier expansion of the weight 4 form (eta(3z))^8 for gamma_0(9). A comparison of the coefficients of x^n for small n gives the identities C=FG-27H^2, and G^2=FH. So mod 3, C^2=G^2=H, and the coefficient of x^p in C^2, when p is a prime congruent to 2 mod 3 is, modulo 3, equal to (1/3)(sigma_1(p))=(p+1)/3. Now the cube of C^2 is the square of f(x^3), where f is the Fourier expansion of delta. It follows that mod 3, the coefficient of x^p in f^2 is (p+1)/3 when p is a prime congruent to 2 mod 3. This is precisely 1. above.