Explicit basis for A//A2 (tmf)

Determine an explicit basis for the Steenrod algebra quotient A//A2, which equals the mod 2 cohomology H^*(tmf) of the connective spectrum for topological modular forms, in a form that enables practical computation analogous to the bases established for H^*(bo) and H^*(bu).

Background

The paper establishes simple admissible-sequence bases for H^*(bo) = A//A1 and H^*(bu) = A//E1 that avoid indices of the form 2^n+1. An attempted generalization to A//An was shown false already for n=2 in degree 49, indicating that a straightforward extension does not work.

The authors state that, although their initial interest was in tmf with H^*(tmf) = A//A2, they do not have a conjecture for A//A2. This leaves open the problem of identifying a tractable basis or structural description for A//A2 comparable in usability to their results for bo and bu.