# Ticket #6393(closed defect: fixed)

Opened 4 years ago

## [with patch, positive review] Jacobi sums incorrect when exactly one character is trivial

Reported by: Owned by: wdj craigcitro major sage-4.1.1 modular forms David Loeffler David Joyner sage-4.1.1.alpha0

### Description

Example:

```sage: G=DirichletGroup(5); X=G.list(); Y=X[0]; Z=X[1]
sage: # Y is trivial and Z is quartic
sage: sum([Y(x)*Z(1-x) for x in IntegerModRing(5)])
-1
sage: # The value -1 above is the correct value of the Jacobi sum J(Y, Z).
sage: Y.jacobi_sum(Z);    Z.jacobi_sum(Y)
0
0
sage: #The 0 values above are incorrect values of J(Y, Z).
```

## Change History

### Changed 4 years ago by wdj

applies to 4.0.2.rc3 passes sage -testall

### comment:1 Changed 4 years ago by wdj

• Owner changed from tbd to craigcitro
• Summary changed from Jacobi sums incorrect when exactly one chacater is trivial to [with patch, needs review] Jacobi sums incorrect when exactly one character is trivial
• Component changed from algebra to modular forms
• Milestone set to sage-4.1

### comment:2 Changed 4 years ago by davidloeffler

• Reviewers set to David Loeffler
• Summary changed from [with patch, needs review] Jacobi sums incorrect when exactly one character is trivial to [with patch, positive review] Jacobi sums incorrect when exactly one character is trivial
• Authors set to David Joyner

Looks good to me, and doctests pass. I'm surprised this went so long without being spotted.

### comment:3 Changed 4 years ago by davidloeffler

BTW, in the course of reviewing this I've realised that our Jacobi sum algorithm is absurdly slow, vastly slower than calculating the sums directly! I've just opened #6534 for this.

### comment:4 Changed 4 years ago by davidloeffler

I'm about to upload a patch at #6534 which totally changes the way we calculate Jacobi sums, thus superseding this patch somewhat. I've made sure that it takes into account the issue you raised here.

### comment:5 Changed 4 years ago by mvngu

Just to let people know, this has been merged in sage-4.1.1-alpha0. I can't close this ticket because I don't have the privilege to do so. Sorry, folks :-(

### comment:6 Changed 4 years ago by mvngu

• Status changed from new to closed
• Resolution set to fixed
• Merged in set to sage-4.1.1.alpha0

### comment:7 Changed 4 years ago by mvngu

See #6613 for a follow up to this ticket.

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