In ticket 10506, John Cremona wrote the following in reference to
Gamma(n): "Note that the next job is to add a method to return a set
of inequivalent cusps. The default implementation is stupidly slow (as
proved by the fact that the old default for ncusps() was to find all
the cusps and count them
Your product formula is a good idea. It's faster than my summation
formula.
On Dec 21, 3:40 am, John Cremona wrote:
> On Dec 21, 1:38 am, rje wrote:
>
> > Thanks for the helpful response. The appropriate code for computing
> > Gamma(n).ncusps() is
>
>
turn ZZ(sum([phi(d)*phi(n/d)/ZZ(2) for d in n.divisors()]))
>
> Why don't you open a ticket to improve this by implementing a suitabel
> formula for the principal congruence subgroups?
>
> John Cremona
>
> On Dec 19, 10:51 pm, rje wrote:
>
>
>
> > Sage is slo
Sage is slow in computing the number of
cusps for Gamma(n).
Look, for example, at the disparity in times below.
sage: time Gamma(5).ncusps()
CPU times: user 52.02 s, sys: 0.24 s, total: 52.26 s
Wall time: 52.29 s
12
sage: time Gamma0(5).ncusps()
CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
s Explorer. So what are these animals? I leave with the
following challenge: Use Sage to determine (algebraically, not just
numerically) the first ten coefficients c(p) of the "eigenform"
corresponding to item 1 in the list.rje
p.s. Just to make sure there is at least one thing useful
What is going on here? Does this only work for even weights? rje
sage: n=numerical_eigenforms(15,3);n.ap(2)
[]
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;s because each term of the
defining sum is a Gaussian integer. For example:
sage: parent(Z(3))
Cyclotomic Field of order 4 and degree 2
rje
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)
0
0
#The 0 values above are incorrect values of J(Y, Z).
Remark: Since unlike Gauss sums mod p, Jacobi sums mod p never
involve p-th roots of unity, the following is also a bit curious:
sage: parent(Z.jacobi_sum(Z))
Cyclotomic Field of order 20 and deg
hardware ?
sage: G=DirichletGroup(18900, GF(193));X=G.list();Y=X[0];
sage: M=ModularSymbols(Y,4,sign=1);
sage: A=(M.T(19)-162).kernel()
rje
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sage: X=DirichletGroup(6).list();
sage: X[0]
[1, 1]
sage: X[0]<>[1, 1]
True
--
Why is this True, and what's the corrected syntax?
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To
sage: X=DirichletGroup(6).list();
sage: X[0]
[1, 1]
sage: X[0]<>[1, 1]
True
--
Why is this True, and what's the corrected syntax?
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To
stion is this: If instead of searching as
above for cases where
c(13) = 14, all I need to find is a case where c(13) = 0 mod 7, is
there a *faster* sage implementation, e.g., using a base field GF(7)?
Thanks. rje
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Transcript below. Ironically, no error is produced by the related
command
sage: time f=D[6].q_eigenform(7,'a')
rje
*
rev...@sobol
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