The following questions arise from modular form theory. But this theory isn't needed to formulate or understand them, and I'm not using the modular-forms tag.

NOTATION

Fix an odd prime $N$. Let $$ F=\sum_{n\text{ an odd square}}x^n\in\mathbb{Z}/2[[x]]. $$ Set $G=F(x^N)$. There is a degree $N+1$ irreducible polynomial relation between $F$ and $G$ over $\mathbb{Z}/2$, (The relation has the form $(F+G)^{N+1}+(\text{lower degree terms}) =0$, and is symmetric).

Examples: Let $S=F+G$ and $P=FG$. When $N=3$, $S^4=P$. When $N=5$, $S^6=P$. When $N=7$, $S^8=P^2+P$. When $N=11$, $(S^4+P)^3=P$.

Now let $M$ be the integral closure of $\mathbb{Z}/2[G]$ in the degree $N+1$ extension field of $\mathbb{Z}/2(G)$ generated by $F$. View $M$ as a subring of $\mathbb{Z}/2[[x]]$, and let $M(\text{odd})$ consist of those $g$ in $M$ for which each exponent $n$ appearing in $g$ is odd. The trace map $\mathbb{Z}/2(F,G)\to \mathbb{Z}/2(G)$ maps the $\mathbb{Z}/2[G^2]$ module $M(\text{odd})$ into the cyclic module generated by $G$. Let $M_0$ consist of all elements of $M(\text{odd})$ of trace 0. $M_0$ is a free rank $N$ module over $\mathbb{Z}/2[G^2]$.

SOME REMARKABLE FACTS

For small $N$ there are very nice bases of $M(\text{odd})$ and $M_0$ over $\mathbb{Z}/2[G^2]$.

$N=3$... Let $C_1=F$, $C_3=G$, $C_5=(F^2)(G)$. Then the $C_j$ are a basis of $M_0$. There is an element of $M(\text{odd})$ whose trace is $G$; it follows that this element and the $C_j$ form a basis of $M(\text{odd})$. Furthermore for each exponent $n$ appearing in $C_j$, $n\equiv j \mod{8}$ and the Legendre symbol $(n/3)$ is either $(j/3)$ or 0.

$N=5$... Let $C_1=F$, $C_3=(F^3)+(G^2)(F)$, $C_5=G$, $C_7=(F^2)(G)$, $C_9=(F^4)(G)$. Then the above results continue to hold, with 3 replaced by 5 in the final sentence.

When $N=7$ or 11 one can write down $C_j$ where $j$ is odd and $<2N$, and prove the corresponding facts. But when $N=11$ it's not possible for all the $C_j$ to be in $Z/2[F,G]$; nevertheless one can arrange that the product of each $C_j$ by $(1+G^8)$ is in $Z/2[F,G]$.

QUESTIONS

To what extent do the above results generalize to larger $N$? More precisely, let $C(\text{plus})$, (resp. $C(\text{minus})$), consist of those $g$ in $M(\text{odd})$ in which each exponent $n$ that appears has $(n/N)$ equal to 0 or 1 (resp. -1). Let $C$ be $C(\text{plus})+C(\text{minus})$.

Question 1--- Does $M_0=C$?

Remark 1... $M$ is indeed the Serre Swinnerton-Dyer ring of characteristic 2 modular power series for $\Gamma_0(N)$, though this isn't obvious. This allows one to introduce Hecke operators.

Remark 2... By using these operators one can show that $C$ has rank $N$ or $N+1$. The bases that I've exhibited when $N=3,5,7$ or 11 show that in each of these cases $M_0$ is contained in $C$. When $N=3,5$ or 7, I can show that $M_0=C$. This is surely also true when $N=11$, though I haven't proved it.

Remark 3... For each odd $n$ prime to $N$ there is a formal Hecke operator $\mathbb{Z}/2[[x]]\to\mathbb{Z}/2[[x]]$. As I've indicated, Remark 1 can be used to show that the $T_n$ stabilize $M$, $M(\text{odd})$ and $C$. So one may ask the perhaps more accessible weakening of Question 1:

Question 2... Do the $T_n$ stabilize $M_0$?