Opened 12 years ago

Closed 12 years ago

# [with patch; with positive review] Atkin-Lehner operator doesn't square to 1

Reported by: Owned by: roed craigcitro major sage-3.0.4 modular forms modular symbols, atkin-lehner

### Description

The following should produce the identity matrix:

sage: e = (DirichletGroup(13).0)^2
sage: M = ModularSymbols(e, 2)
sage: M.atkin_lehner_operator().matrix()^2
[         1          0          0          0]
[         0          1          0          0]
[-zeta6 - 1          0          1  zeta6 + 1]
[ zeta6 + 1          0          0     -zeta6]


### comment:1 Changed 12 years ago by was

For me this illustrates the bug more clearly:

sage: M = ModularSymbols(Gamma1(13),2)
sage: M
Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Rational Field
sage: M.atkin_lehner_operator(13).matrix()^2 == 1
True
sage: M = ModularSymbols(DirichletGroup(13).0^2)
sage: M.atkin_lehner_operator(13).matrix()^2 == 1
False
sage: M.atkin_lehner_operator(13).matrix()^2

[         1          0          0          0]
[         0          1          0          0]
[-zeta6 - 1          0          1  zeta6 + 1]
[ zeta6 + 1          0          0     -zeta6]


### comment:2 Changed 12 years ago by was

• Summary changed from Atkin-Lehner operator doesn't square to 1 to [with patch; needs review] Atkin-Lehner operator doesn't square to 1

WARNING:

The atkin-lehner operator does *not* leave the space $M_k(N,\chi)$ invariant unless $\chi$ is quadratic. Really it sends $M_k(N,\chi)$ to $M_k(N,\chibar)$. So Sage should give an error message when $\chi$ is not quadratic.

### comment:3 Changed 12 years ago by craigcitro

• Summary changed from [with patch; needs review] Atkin-Lehner operator doesn't square to 1 to [with patch; with positive review] Atkin-Lehner operator doesn't square to 1

Looks good. I added a patch that actually corrects a bug (some statements were out of order), and just reformats/corrects typos. This is ready to go.

### comment:4 Changed 12 years ago by mabshoff

• Milestone changed from sage-3.1.1 to sage-3.0.4
• Resolution set to fixed
• Status changed from new to closed

Merged in Sage 3.0.4.alpha2

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