The space of mod-3 q-expansions of modular forms of level one and any weight is isomorphic to M_{3} = F_{3}[Δ], where Δ = q + q^{4} + 2q^{7} + 2q^{13} + q^{16} + ⋯ is the normalized cuspform of weight 12. Since the prime-to-3 Hecke operators all commute with the cubing map, to understand the Hecke action on M_{3}, it suffices to understand the action on the subspace of M_{3} spanned by the prime-to-3 powers of Δ. This subspace K_{3} = F_{3}⟨Δ, Δ^{2}, Δ^{4}, Δ^{5}, ... ⟩ is the kernel of the U_{3} operator.
One can prove that the completed Hecke algebra generated by the action of the Hecke operators T_{n} with n prime to 3 on K_{3} is isomorphic to F_{3}[[T_{2}, 1 + T_{7}]]. (For details, see quick sketch here, original outline of proof in the appendix to Bellaïche-Khare, or full argument in section 8.3 of my thesis.) By duality, there is a unique basis {m(a, b)}_{a ≥ 0, b ≥ 0} of K_{3} adapted to x = T_{2} and y = 1 + T_{7}, in the sense that
Values of m(a, b) for a + b ≤ 17, presented as polynomials in y = Δ over F_{2}. Computed by linear algebra in SAGE.
The computations here are inspired by Nicolas and Serre's analysis of the case p = 2. They study the F_{2}-vector space spanned by odd powers of Δ: this space K_{2} is the analogue of K_{3} for modular forms modulo 2. In particular, they prove that the Hecke algebra generated by the action of odd Hecke operators on K_{2} is F_{2}[[T_{3}, T_{5}]]. Therefore there is a basis of K_{2} adapted to x = T_{3} and y = T_{7} as above; they compute coefficients of elements of this basis.