A Hecke algebra attached to mod 2 modular forms of level 3

Let $D$ in $Z\mathrm{/}2[[x]]$ be $\sum x^{n^2}$, $n>0$ and prime to $6$. Let $W$ be spanned by the $D^k$, $k>0$ and prime to $6$. Then the formal Hecke operators $T_p$, $p>3$, stabilize $W$, and it can be shown that they act locally nilpotently. We show that the completion of the Hecke algebra generated by these $T_p$ acting on $W$, with respect to the maximal ideal generated by the $T_p$, is a power series ring in $T_7$ and $T_{13}$ with an element of square $0$ adjoined. This may be viewed as a level 3 analog of the level 1 results of Nicolas and Serre – the Hecke stable space they study is spanned by the odd powers of the mod $2$ reduction of $\mathrm{\Delta }$, and their resulting completed Hecke algebra is a power series ring in $T_3$ and $T_5$.