2 Change $\chi_3$ to $\chi$ for consistency

The formula that emiliocba seeks seems to be as follows. Let $\chi_3$ \chi$be the Dirichlet character mod$3$. For$k>0$write$k = 3^e n$with$n \equiv \pm 1 \bmod 3$. Then the number of representations of$k$by this quadratic form$A_2^3$is $$s(k) := 9 (3^{2e+1}-\chi(n)) \phantom. \sum_{d|n} \phantom. \chi(n/d)\phantom. d^2.$$ I append gp code that verifies that this holds for$k \leq 432$. To prove it in general it will be enough to check that $$\varphi := 1 + \sum_{k=1}^\infty \phantom. s(k) q^k$$ is a modular form of weight$3$and character$\chi$for$\Gamma_0(3)$, and to match a few coefficients with the theta function$\theta_{A_2^3}$. In principle, it is enough to match only the$q^0$coefficient: the dual of$A_2^3$is isomorphic with the scaling of$A_2^3$by$1/3$, so by Poisson summation$\theta_{A_2^3}$is modular also for the normalizer$\Gamma_0^+(3)$of$\Gamma_0(3)$(generated by$\Gamma_0(3)$and the involution$w_3 : \tau \longleftrightarrow -1/3\tau\phantom.$); and$\Gamma_0^+(3)$has only one cusp, and no cusp forms of weight less than$6$(the weight of$\eta(\tau)^6 \eta(3\tau)^6$), so the normalized Eisenstein series$\varphi$is the only candidate for$\theta_{A_2^3}$. H = 24 A2 = sum(m=-H,H,sum(n=-H,H,q^(m^2+m*n+n^2))) + O(q^(3*H^2/4+1)); L = A2^3; chi3(m) = kronecker(m,3) { s(k, e,n) = e = valuation(k,3); n = k / 3^e; 9 * (3^(2*e+1)-chi3(n)) * sumdiv(n, d, chi3(n/d)*d^2) } L == 1 + sum(k=1,3*H^2/4,s(k)*q^k)  1 The formula that emiliocba seeks seems to be as follows. Let$\chi_3$be the Dirichlet character mod$3$. For$k>0$write$k = 3^e n$with$n \equiv \pm 1 \bmod 3$. Then the number of representations of$k$by this quadratic form$A_2^3$is $$s(k) := 9 (3^{2e+1}-\chi(n)) \phantom. \sum_{d|n} \phantom. \chi(n/d)\phantom. d^2.$$ I append gp code that verifies that this holds for$k \leq 432$. To prove it in general it will be enough to check that $$\varphi := 1 + \sum_{k=1}^\infty \phantom. s(k) q^k$$ is a modular form of weight$3$and character$\chi$for$\Gamma_0(3)$, and to match a few coefficients with the theta function$\theta_{A_2^3}$. In principle, it is enough to match only the$q^0$coefficient: the dual of$A_2^3$is isomorphic with the scaling of$A_2^3$by$1/3$, so by Poisson summation$\theta_{A_2^3}$is modular also for the normalizer$\Gamma_0^+(3)$of$\Gamma_0(3)$(generated by$\Gamma_0(3)$and the involution$w_3 : \tau \longleftrightarrow -1/3\tau\phantom.$); and$\Gamma_0^+(3)$has only one cusp, and no cusp forms of weight less than$6$(the weight of$\eta(\tau)^6 \eta(3\tau)^6$), so the normalized Eisenstein series$\varphi$is the only candidate for$\theta_{A_2^3}\$.

H = 24
A2 = sum(m=-H,H,sum(n=-H,H,q^(m^2+m*n+n^2))) + O(q^(3*H^2/4+1));
L = A2^3;

chi3(m) = kronecker(m,3)
{
s(k, e,n) =
e = valuation(k,3);
n = k / 3^e;
9 * (3^(2*e+1)-chi3(n)) * sumdiv(n, d, chi3(n/d)*d^2)
}

L == 1 + sum(k=1,3*H^2/4,s(k)*q^k)