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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(783, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([7,117]))
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pari:[g,chi] = znchar(Mod(515,783))
| Modulus: | \(783\) | |
| Conductor: | \(783\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(126\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
|
| Real: | no |
| Primitive: | yes |
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sage:chi.is_primitive()
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pari:#znconreyconductor(g,chi)==1
|
| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{783}(5,\cdot)\)
\(\chi_{783}(38,\cdot)\)
\(\chi_{783}(92,\cdot)\)
\(\chi_{783}(122,\cdot)\)
\(\chi_{783}(149,\cdot)\)
\(\chi_{783}(158,\cdot)\)
\(\chi_{783}(167,\cdot)\)
\(\chi_{783}(209,\cdot)\)
\(\chi_{783}(212,\cdot)\)
\(\chi_{783}(236,\cdot)\)
\(\chi_{783}(245,\cdot)\)
\(\chi_{783}(254,\cdot)\)
\(\chi_{783}(266,\cdot)\)
\(\chi_{783}(299,\cdot)\)
\(\chi_{783}(353,\cdot)\)
\(\chi_{783}(383,\cdot)\)
\(\chi_{783}(410,\cdot)\)
\(\chi_{783}(419,\cdot)\)
\(\chi_{783}(428,\cdot)\)
\(\chi_{783}(470,\cdot)\)
\(\chi_{783}(473,\cdot)\)
\(\chi_{783}(497,\cdot)\)
\(\chi_{783}(506,\cdot)\)
\(\chi_{783}(515,\cdot)\)
\(\chi_{783}(527,\cdot)\)
\(\chi_{783}(560,\cdot)\)
\(\chi_{783}(614,\cdot)\)
\(\chi_{783}(644,\cdot)\)
\(\chi_{783}(671,\cdot)\)
\(\chi_{783}(680,\cdot)\)
...
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((407,379)\) → \((e\left(\frac{1}{18}\right),e\left(\frac{13}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 783 }(515, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{89}{126}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{59}{63}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
