The space $P1$ of my earlier question 203755 "Two spaces attached to mod 2 level 9 modular forms...", is essentially the space of mod 2 level 3 modular forms. That such a space should appear inside the space of level 9 forms is evident, but that it appears in a different position in this space, in a different guise, as $P2/P1$, seems to me a mystery. Here's a related mystery in level 25. I'm quite certain of my conjecture, and even how to prove it, but would be grateful for any insight as to why it's true.

**NOTATION**

As in the level 9 case, $F$ in $Z/2[[x]]$ is $x+x^9+x^{25}+x^{49}+\ldots$, but now $G=F(x^5)$, $D=x+x^9+x^{49}+x^{81}+\ldots$ and $E=x+x^4+x^9+x^{16}+x^{36}+\ldots$ where the exponents in $D$ and $E$ are the squares prime to 10 and 5 respectively. $S$ and $T$ are $(E^{16})*G$ and $(E^8)*G$.

$Z/2[G^2]$ is a subring of $Z/2[[x]$]; view the latter as a module over the former. Define submodules $P1$ and $P2$ of $Z/2[[x]]$ of ranks 4 and 6 as follows: A basis of $P1$ consists of $D1=D$, $D3=(D^8)/G$, $D7=(D^2)*G$ and $D9=(D^4)*G$. P2 is generated by $P1$, $S$, and $T$. It can be shown, using modular forms of levels 5 and 25, that $P1$ and $P2$ are "Hecke-stable", i.e. that they're stabilized by the $T_p$ with $p$ not 2 or 5. Let $H=F(x^{25})$.

**CONJECTURE**

Let $P3$ be the $Z/2[G^2]$-module of rank 10 generated by $ P2, Y1, Y3, Y7$, and $Y9$, where $Y9=(D^2)*(G)*(G+H)^2$, $Y7=T_7(Y9)$, $Y3=T_3(Y9)$, and $Y1=T_3(Y3)$. Then:

(a) $P3$ is Hecke-stable

(b) The $Z/2[G^2]$-linear isomorphism $P1\to P3/P2$ that takes $Di$ to $Yi$, $i=1,3,7,9$ preserves the Hecke action.

**NOTE**

There is as far as I can see no a priori reason for believing (a) or (b). But the computer insists that they're true and I'm sure that an ad hoc proof, along the lines I sketched in my earlier question, can be cobbled out.

**QUESTION**

What lies behind the above isomorphism (and the similar isomorphism in level 9)?

EDIT--A FURTHER MIRACLE

Let P be the subspace of Z/2[[x]] consisting of mod 2 modular forms f for Gamma_0 (25) such that all exponents appearing in f are prime to 10. P is contained in Z/2(F,G,H) where H is G(x^5). If f lies in P one writes f as f(plus)+f(minus) where all exponents k appearing in f(plus) (resp. f(minus)) are squares (resp. non-squares) mod 5. f(plus) and f(minus) lie in P.

Let P# consist of those f in P such that the traces of f(plus) and f(minus) from Z/2(F,G,H) to Z/2(F,G), i.e. from level 25 to level 5, are both 0. P# is Hecke-stable and stable under multiplication by G^2. As a Z/2[G^2] module it has rank 18.

Now there are two mod 2 eigensystems of level 25, the first attached to delta, and the second to a weight 4 level 25 newform. Accordingly P# is a direct sum of two generalized eigenspaces.I see no reason why either of these spaces should be stable under multiplication by G^2. But now the miracle occurs.

The power series D1, D3, D7, D9, S, T, Y1, Y3, Y7, and Y9 that I introduced in my question all lie in P#. So the rank 10 Z/2[G^2] module P3 is contained in P#, and using its Hecke structure one finds that it is a subspace of the first generalized eigenspace.

I'm now convinced that I can write down 8 elements of the second generalized eigenspace that generate a complement to P3 in P#, and show that this complement is Hecke-stable and contained in the second generalized subspace.

So I can decompose P# explicitly, and show that P3 is in fact the entire first generalized eigenspace.This "explains" why P3 is Hecke-stable, only to raise further questions. Why does P3 have its peculiar Hecke-module structure? Why are the generalized eigenspaces in this situation stable under multiplication by G^2? And does any of this curious level 9 and level 25 theory generalize to other levels and characteristics?

EDIT___THE LEVEL 27 SITUATION

The analogous situation in level 27 is illuminating, though it requires a bit of preliminary notation. As usual, F in Z/2[[x]] is x+x^9+x^25+ ... , the exponents being the odd squares. Now G,H and J will be F(x^3), F(x^9) and F(x^27). F,G,H and J are the mod 2 reductions of delta(z), delta(3z), delta(9z) and delta(27z). It turns out that one wants to view the space M(27,odd) of odd mod 2 modular forms of level Gamma_0 (27) as a module over Z/2[(G+H)^2]; the point being that the Fricke involution W_27 fixes G+H. As such a module, M(27,odd) is free of rank 18. Though I haven't written out full details I believe I can show:

M(odd,27) is a direct sum of 3 Hecke stable submodules of ranks 4, 4, and 10. These are the generalized eigenspaces corresponding to the three mod 2 eigensystems of level Gamma_0 (27). (Alex Ghitza has shown that there are only three mod 2 eigensystems of this level).

One first wants to work with M(27,odd,small), the rank 9 Hecke stable submodule of M(27,odd) consisting of elements fixed by W_27. This has a decomposition into Hecke stable submodules of ranks 2,2, and 5, and it shouldn't be hard to get from this result to the result for the full space M(27,odd).

Explicitly one may construct mod 2 modular forms t and B of level 27 with the following properties:

a) t is the reduction of the level 27 weight 2 cusp form, and t^3 = G+H.

b) t*(B^4) is the reduction of a certain level 108 weight 4 newform, and 1+(B^3) = t*B.

c) A Z/2[t^6]- basis of M(27,odd,small) consists of t, t^5, t* (B^4), (t^5)*(B^2),

t^3, (t^3)* (B^2), (t^3)* (B^4), t* (B^2), and (t^5)* (B^4).

d) The first 2 elements in this basis generate the generalized eigenspace corresponding to the eigensystem attached to t. The third and fourth elements generate the generalized eigenspace corresponding to the eigensystem attached to t*(B^4). The last 5 elements generate the eigenspace corresponding to the level 1 eigensystem.

Finally I give precise descriptions of t and B as power series and an outline of the argument. t is F + J + (F*J)^ (1/4). B is 1+ E(x)+ E(x^3), where E is the level 9 modular form x+x^4+x^16+x^25 ... , the exponents being the squares prime to 3. Since W_27 interchanges F and J and also interchanges E(x) and 1+E(x^3),it fixes t and B.

Using recursions attached to T_5 and T_7 one shows that the submodule generated by t and t^5 is stabilized by T_5 and T_7 + I, and that these operators are degree decreasing. Results of a similar sort hold for the second submodule (and the operators T_5 + I and T_7 + I) and for the third submodule (and the operators T_5 and T_7). This gives the stability of the three submodules under the T_p for all p>3, and since there are only 3 generalized mod 2 eigensystems of level 27, our spaces are just the generalized eigenspaces attached to these eigensystems.