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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 $Z/2[t]$ defined by a linear recurrence and showed that each $C_{n}$, $n\equiv 0 \bmod{4}$, 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 $Γ_{0}(3)$ 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 $Γ_{0}(5)$. 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 $Γ_{0}(3)$ results from 1508.07523, we use level $Γ_{0}(5)$ results from arXiv:1603.07085 [math.NT]. A linear recurrence again plays the key role. Now $C_{n+6} = C_{n+5} + (t^{6}+t^{5}+t^{2}+t)C_{n}+t^{n}(t^{2}+t)$, $C_{0}=0$, $C_{1}=C_{2}=1$, $C_{3}=t$, $C_{4}=t^{2}$, $C_{5}=t^{4}+t^{2}+t$, and we prove that each $C_{n}$, $n\equiv 0$ or $2\bmod{6}$ is a sum of $C_{k}$, $k<n$.