A characteristic 2 recurrence related to $U_5$, with a Hecke algebra application

In arXiv:1603.03910 [math.NT] we introduced some $C_n$ in $Z\mathrm{/}2[t]$ defined by a linear recurrence and showed that each $C_n$, $n\equiv 04$, is a sum of $C_k$, $k<n$. Combining this with results from arXiv:1508.07523 [math.NT] we proved that the space $K$, consisting of those odd mod~2 modular forms of level ${\mathrm{\Gamma }}_0(3)$ that are annihilated by the operator $U_3+I$, has a basis $m_{i,j}$ “adapted to $T_7$ and $T_{13}$” in the sense of Nicolas and Serre. (And so the “completed shallow Hecke algebra” attached to $K$ is a power series ring in $T_7$ and $T_{13}$.) This note derives analogous results in level ${\mathrm{\Gamma }}_0(5)$. Now $U_3+I$ is replaced by $U_5+I$, and the operators $T_7$ and $T_{13}$ by $T_3$ and $T_7$. In place of level ${\mathrm{\Gamma }}_0(3)$ results from 1508.07523, we use level ${\mathrm{\Gamma }}_0(5)$ results from arXiv:1603.07085 [math.NT]. A linear recurrence again plays the key role. Now $C_{n+6} = C_{n+5} + (t^{6}+t^{5}+t^{2}+t)C_{n}+t^{n}(t^{2}+t)$,$C_{0}=0$,$C_{1}=C_{2}=1$,$C_{3}=t$,$C_{4}=t^{2}$,$C_{5}=t^{4}+t^{2}+t$,andweprovethateach$C_{n}$,$n\equiv 0$or$2\bmod{6}$isasumof$C_{k}$,$k<n$.