How explain these remarkable empirical observations about mod 3 modular forms of levels 1 and 5?

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 o.e.i.s but the similar Moser-de Bruijn sequence 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 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 T2: Z/3[[q]]–> Z/3[[q]] stabilizes V. (V is the space of level 1 mod 3 modular forms killed by U3). Let K be the kernel of T2: V–> V, and s(0) < s(1) < s(2) ... 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 Z/3[F], n prime to 3, be T2(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 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 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 U3. Let V be the subspace of Z/3[t] spanned by the t^k, k prime to 3. Let i: V–> Z/3[[q]] be the 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 Tp, 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 U3 and fixed by the mod 3 Fricke involution W5). Let K be the kernel of (T2 + I): i(V)–> i(V). Let t(0) < t(1) < t(2) .. 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 T7.

CONJECTURE 4: The conjecture still holds when K is replaced by the kernel of T11 + 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 (T7 _ 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.

All topic

discuss about the differentiability of \$g(x)=|f(x)|\$, where \$f\$ is a differentiable function

I want to discuss about the differentiability of \$g(x)=|f(x)|\$, where \$f\$ is a differentiable function

Example 1

Take \$f(x)=|x|\$, function is clearly not differentiable at \$x=0\$.

Example 2

Take \$f(x)=|sin(x)|\$, function is clearly not differentiable at the point \$x=npi\$

After taking few more examples like \$|x-1|, |cos(x)|\$, it always seems to be the case that \$|f(x)|\$ is not differentiable at the points where \$f(x)=0\$

Observation: One thing is common in all the examples that some portion of \$f(x)\$ lies below \$x\$ axis.
So I took another example

\$f(x)=x^2\$ but \$|f(x)|\$ is differentiable at the point where \$f(x)=0\$

Question 1:

Am I right in concluding that we can not just say in general setting that \$|f(x)|\$ is not differentiable at the points where \$f(x)=0\$?

When can we(I mean under what conditions can we )conclude that \$|f(x)|\$ is differentiable at points where \$f(x)=0\$. My hypothesis is that graph of \$f\$ should lie below \$x\$ axis.

Question 2:

Let \$f(x)\$ and \$g(x)\$ be two differentiable function, when can we conclude that \$|f(x)|+|g(x)|\$ is not differentiable at the points where \$f(x)=0\$ and \$g(x)=0\$

Example \$|sin(2-x)|+ |cos(x)| \$ are not differentiable at \$x=2+2pi, x=(2n+1)frac{k}{2}\$

All topic

Why do people not talk about Tokyo fire bombing during WW2? [on hold]

I was wondering why Hiroshima and Nagasaki are remembered as some of the most devastating attacks during WW2 but the Tokyo firebombing March 9, 1945 which caused more casualties and was a really devastating blow on the nations capital but it has been largely forgotten in modern times why is that?

please keep in mind I am not an expert on the subject and I’m not sure about the exact number of casualties and injuries caused by the fire bombings.

All topic

Looking for a 80s Novel Series about post apocalypse world where 4 soldiers try and bring order

Looking for a novel series I read in the 1980s about 4 soldiers (the leader was a US Marine Force Recon member). There was also a Navy Officer, Green Beret soldier and Air Force fighter pilot.
They were some sort of special team put together by the last president before the war…and now travel around trying to restore order to the US after a War (Nuclear War I think).
Any help would be appreciated.
Thank you,
Robert

All topic

Is there anything uniquely difficult about repealing Treaty law?

The proposed “Chequers Agreement” if passed would involve a new treaty committing the UK to ongoing regulatory alignment for goods.

Some Leavers view this with suspicion because they view this kind of alignment as difficult to reverse.

Is this because Treaty law is uniquely difficult to repeal? Or is it more that they do not believe the political will to repeal it would ever be present?

All topic

Horror movie about prisoners encountering cannibals family [on hold]

It is about a few prisoners that are transported in a small van in to the country side, they managed to let the van stop and kill the cops with a big rock on his head (you see a dent in the head I remember)

After this they find a small wooden house in the mountains on a grassy area (not in dense woods). There is a Christian family in the house that are cannibals and try to eat/kill the prisoners. They had a cross above their door. They invited the prisoners in and made them comfortable, but turned cannibalistic on them.

All topic

Horror movie about prisoners encountering cannibals family

It is about a few prisoners that are transported in a small van in to the country side, they managed to let the van stop and kill the cops with a big rock on his head (you see a dent in the head I remember)

After this they find a small wooden house in the mountains on a grassy area (not in dense woods). There is a Christian family in the house that are cannibals and try to eat/kill the prisoners. They had a cross above their door. They invited the prisoners in and made them comfortable, but turned cannibalistic on them.

All topic

Macbooks randomly disconnecting from wifi about twice a day [on hold]

I kind of placed into a network admin position due to my proximity of the hardware and very slight knowledge of the subject at a growing company. One of our offices is in a super crowded area and all our neighbors seem to have their own wifi + broadcasting a wonderful xfinity signal as well. I’m guessing this is my main issue, but all our macbooks seem to randomly disconnect once or twice a day. It happens for about a minute, and sometimes requires turning the wireless on the mac off and on.

I’ve got a sonicwall and a couple of ruckus r610s. I previously had some ubiquiti WAPs in there but replaced them hoping the ruckus waps would perform better (they haven’t so far…). Anyone have any advice or experience with something similar?

Things ive tried: Running new cable, replacing WAPs, trying different channels on waps, trying different ruckus firmware, manually setting MTU on mac.

Things I should probably try: Going to each of our 20~ neighbors and asking them to turn off their xfinity wifi.

All topic

Please recommend papers/books/blogs for someone trying to learn about the Holographic Principle, assuming that person has knowledge of the undergraduate mathematics and physics major level.

All topic

Question about Fermat’s Last Theorem’s proof (Wiles’s proof)

Question -1 What level of math do I need to understand the proof Andrew Wiles wrote? Am I supposed to be a mathematics professor? For example, I don’t understand anything from these pages. That’s a really bad feeling.

Question-2 This question may not look nice. It’s ridiculous to ask this question. One of our teachers said in the course: “Andrew Wiles’s proof was not actually approved. Only the mathematicians accepted this as true.” Of course, I don’t believe it. But there was a doubt in me.

Anyway, my main question is the first question I ask.

For example, I don’t know anything about these mathematical notations:

Thank you very much.

All topic