Jacobi sums incorrect when exactly one character is trivial

[wdjoyner]

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).

Component: modular forms

Author: David Joyner

Reviewer: David Loeffler

Merged: sage-4.1.1.alpha0

Issue created by migration from https://trac.sagemath.org/ticket/6393

[wdjoyner]

Attachment: trac_6393-jacobi-sum.patch.gz

applies to 4.0.2.rc3 passes sage -testall

[loefflerd]

Reviewer: David Loeffler

[loefflerd]

Author: David Joyner

[loefflerd]

comment:2

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

[loefflerd]

comment:3

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.

[loefflerd]

comment:4

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.

[sagetrac-mvngu]

comment:5

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 :-(

[sagetrac-mvngu]

Merged: sage-4.1.1.alpha0

[sagetrac-mvngu]

comment:7

See #6613 for a follow up to this ticket.