Define $b(n)$ by $b(0)= 0$, $b(3n)= 9b(n)$, $b(3n+1)= 9b(n) + 1$, $b(3n+2)= 9b(n) + 3$. (This sequence does not show up in the OEIS, but the similar Moser-de Bruijn sequence [A000695](https://oeis.org/A000695) appears in many situations, one being the theory of mod 2 modular forms).

One seems to encounter the $b(n)$ when studying mod 3 modular forms of level $\Gamma_0 (N)$ for various small $N$. But what follows is largely empirical though supported by overwhelming evidence. I would welcome (but do not expect, particularly in the mysterious level 5 case) an explanation.

##LEVEL 1##

Let $F$ in $\mathbf{Z}/3[[q]]$ be the mod 3 reduction of the level 1 weight 12 cusp form. Let $V$ be the subspace spanned by the $F^k$ with $(k,3)= 1$. The formal Hecke operator $T_2: \mathbf{Z}/3[[q]] \to\mathbf{Z}/3[[q]]$ stabilizes $V$. ($V$ is the space of level 1 mod 3 modular forms killed by $U_3$). Let $K$ be the kernel of $T_2: V\to V$, and $s(0) < s(1) < s(2) <\cdots$ the degrees (in $F$) of the (non-zero) elements of $K$ arranged in order of increasing size.

###CONJECTURE 1:   $s(n)= 9b(n) + 1$.###

This has been computer verified for $n < 54$; in particular $s(53)= 9019$. Here's how one makes the calculations. Let $A(n)$ in $\mathbf{Z}/3[F]$, $n$ prime to 3, be $T_2(F^n)$. The $A(n)$ satisfy the degree 9 recursion $A(n+9)= 2(F^9)A(n) + (F^3)A(n+3)$, with initial conditions $A(1)= 0$, $A(2)= F$, $A(4)= F^2$, $A(5)= F^4$, $A(7)= F^5$, $A(8)= F^7 + F^4$. This permits the rapid calculation of the $A(n)$ and of those $n$ for which $A(n)$ is a $\mathbf{Z}/3$-linear combination of $A(k)$ for $k < n$. These $n$ are the degrees of the elements of $K$.

##LEVEL 5##

Let $v$ in $\mathbf{Z}/3[[q]]$ be the mod 3 reduction of the weight 4 cusp form of level $\Gamma_0 (5)$. Then $v(1) = v + v^3$ and $v(2)= v^2 - v^3$ are killed by $U_3$. Let $V$ be the subspace of $\mathbf{Z}/3[t]$ spanned by the $t^k$, $k$ prime to 3. Let $i: V\to \mathbf{Z}/3[[q]]$ be the $\mathbf{Z}/3$-linear imbedding taking $t^{3k+1}$ to $(v^{3k})v(1)$ and $t^{3k+2}$ to $(v^{3k})v(2)$. The identification of $i(V)$ with $V$ gives a degree function (degree in $t$) on $i(V)$. It can be shown that the $T_p$, when $p$ is other than 3 or 5, stabilize $i(V)$. ($i(V)$ is the space of mod 3 modular forms of level $\Gamma_0 (5)$ killed by $U_3$ and fixed by the mod 3 Fricke involution $W_5$). Let $K$ be the kernel of $(T_2 + I): i(V)\to i(V)$. Let $t(0) < t(1) < t(2) <\cdots$ be the degrees of the elements of $K$ arranged in order of increasing size.

###CONJECTURE 2:  $t(2n)= 27b(n) + 1$ or $27b(2n) + 2$ according as $b(n)$ is even or odd. $t(2n+1)= 27b(n) + 11$ or $27b(n) + 10$ according as $b(n)$ is even or odd.###

###CONJECTURE 3: The conjecture still holds when $K$ is replaced by the kernel of $T_7$.###

###CONJECTURE 4: The conjecture still holds when $K$ is replaced by the kernel of $T_{11} + I$.###

All three of these conjectures hold for $n < 27$. In particular all three $t(53)$ are 7381. The recursions used to establish these results are messy; they are of degrees 12, 24, and 39 respectively.

Remark -- There is a similar computer supported conjecture in level 11 for the kernel of $(T_7 - I)$ acting on a space analogous to $i(V)$. But now $v$ is the reduction of the weight 2 cusp form of level $\Gamma_0 (11)$, $v(1)= v + v^3$ and $v(2)= v^2 + v^3 + v^6$.