The space of mod-3 q-expansions of modular forms of level one and any weight is isomorphic to M3 = F3[Δ], where Δ = q + q4 + 2q7 + 2q13 + q16 + ⋯ 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 M3, it suffices to understand the action on the subspace of M3 spanned by the prime-to-3 powers of Δ. This subspace K3 = F3⟨Δ, Δ2, Δ4, Δ5, ... ⟩ is the kernel of the U3 operator.
One can prove that the completed Hecke algebra generated by the action of the Hecke operators Tn with n prime to 3 on K3 is isomorphic to
F3[[T2, 1 + T7]].
(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 K3 adapted to x = T2 and y = 1 + T7, in the sense that
Values of m(a, b) for a + b ≤ 17, presented as polynomials in y = Δ over F2. 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 F2-vector space spanned by odd powers of Δ: this space K2 is the analogue of K3 for modular forms modulo 2. In particular, they prove that the Hecke algebra generated by the action of odd Hecke operators on K2 is F2[[T3, T5]]. Therefore there is a basis of K2 adapted to x = T3 and y = T7 as above; they compute coefficients of elements of this basis.