Fix an odd prime $p$. Each $f \in \mathbb{Z}/2[[x]]$ can be written as $f\_{+} + f\_{-} + f\_0$ where each exponent k of $x$ appearing in $f\_{+}$ (resp. $f\_{-}$, $f\_0$) has Legendre symbol $(k/p)$ equal to 1 (resp. -1, 0).


When $f$ is a "characteristic 2 modular form for $\Gamma\_0(p)$", the same need not hold for $f\_{+}$, $f\_{-}$, and $f\_0$.


QUESTION: When $f$ is a characteristic 2 modular form for $\Gamma\_0(p^2)$, is the same true of $f\_{+}$, $f\_{-}$ and $f\_0$?


Remarks: The answer to the question is yes when $p$ is 3 or 5. One can give a formulation of the question free from any reference to modular forms since the space of characteristic 2 modular forms of level $p^2$ is the integral closure of $\mathbb{Z}/2[F,G,H]$ in its field of fractions,
where $F=x+x^9+x^{25}+x^{49}+...$ and $G=F(x^p)$, $H=G(x^p)$.



A question about decomposing mod 2 modular forms of level p^2
Yes. In fact, if $f$ is a characteristic zero modular form of level $p^{2}$ then $f\_{+}$, $f\_{-}$ and $f\_{0}$ are also characteristic zero modular forms of level $p^{2}$.


We have $f\_{0} = f | U(p) | V(p)$, and since $f$ has level $p^{2}$, $f | U(p)$ has level $p$ and then $f | U(p) | V(p)$ has level $p^{2}$.


Also, since $f = \sum a(n) q^{n}$, $f\_{\chi} = \sum \chi(n) a(n) q^{n}$ has level $p^{2}$ if $\chi$ is the Legendre symbol modulo $p$. This follows, for example, by Theorem 7.4 from Iwaniec's "Topics in Classical Automorphic Forms". Since $f\_{+} = \frac{1}{2} (f - f\_{0}) + \frac{1}{2} f\_{\chi}$, we have that $f\_{+}$ also has level $p^{2}$. Then $f\_{-} = f - f\_{0} - f\_{+}$ also has level $p^{2}$.


