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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:

  1. When k=2, S(1) is all p that are 2 mod 9, S(2) all p that are 5 mod 9

  2. When k=4, S(1) is all p that are 4 or 7 mod 9, S(2) is empty

  3. 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

  4. 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.

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  • $\begingroup$ I think the tag reference-request exists, and that it would be to your advantage to use it. Gerhard "Or Else Suffer The Consequences" Paseman, 2012.09.26 $\endgroup$
    – Gerhard Paseman
    Commented Sep 26, 2012 at 15:54
  • $\begingroup$ $\Delta^k = \eta^{24k}$, which mod 3 is basically $\eta^{8k}$. You might also try powers $\eta^m$ where $m$ is not a multiple of 8. $\endgroup$
    – Noam D. Elkies
    Commented Sep 27, 2012 at 2:11
  • $\begingroup$ It seems to me like a full answer to this question would be the mod 3 analogue of the beautiful work of Serre and Nicolas on modular forms modulo 2: arxiv.org/abs/1204.1039 $\endgroup$
    – Will Sawin
    Commented Dec 13, 2012 at 7:57
  • $\begingroup$ @Will--I agree. Let g in Z/2[[x]] be the characteristic 2 analogue of f. Joel Bellaiche proved a conjecture of Nicolas and Serre, and using this found just which linear combinations of the g^k with k odd corresponded to abelian Galois representations. (In particular the g^k with k=3,5,7,19 and 21 are "abelian"--I write a little about this on other MO questions). I've experimentally confirmed a characteristic 3 analogue of Joel's result on linear combinations, but have no proofs. However my question about f^k where k=2,4,5 or 10 perhaps admits a more elementary answer--(to be continued) $\endgroup$
    – paul Monsky
    Commented Dec 14, 2012 at 12:37
  • $\begingroup$ Ramanujan determined the mod 27 reduction of the Fourier expansion of delta, and perhaps others have worked on my particular powers of delta. $\endgroup$
    – paul Monsky
    Commented Dec 14, 2012 at 12:40

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