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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(783, base_ring=CyclotomicField(252))
M = H._module
chi = DirichletCharacter(H, M([70,9]))
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pari:[g,chi] = znchar(Mod(437,783))
| Modulus: | \(783\) | |
| Conductor: | \(783\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(252\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{783}(2,\cdot)\)
\(\chi_{783}(11,\cdot)\)
\(\chi_{783}(14,\cdot)\)
\(\chi_{783}(32,\cdot)\)
\(\chi_{783}(47,\cdot)\)
\(\chi_{783}(50,\cdot)\)
\(\chi_{783}(56,\cdot)\)
\(\chi_{783}(68,\cdot)\)
\(\chi_{783}(77,\cdot)\)
\(\chi_{783}(95,\cdot)\)
\(\chi_{783}(101,\cdot)\)
\(\chi_{783}(113,\cdot)\)
\(\chi_{783}(119,\cdot)\)
\(\chi_{783}(131,\cdot)\)
\(\chi_{783}(137,\cdot)\)
\(\chi_{783}(155,\cdot)\)
\(\chi_{783}(164,\cdot)\)
\(\chi_{783}(176,\cdot)\)
\(\chi_{783}(182,\cdot)\)
\(\chi_{783}(185,\cdot)\)
\(\chi_{783}(200,\cdot)\)
\(\chi_{783}(218,\cdot)\)
\(\chi_{783}(221,\cdot)\)
\(\chi_{783}(230,\cdot)\)
\(\chi_{783}(263,\cdot)\)
\(\chi_{783}(272,\cdot)\)
\(\chi_{783}(275,\cdot)\)
\(\chi_{783}(293,\cdot)\)
\(\chi_{783}(308,\cdot)\)
\(\chi_{783}(311,\cdot)\)
...
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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 ]
\((407,379)\) → \((e\left(\frac{5}{18}\right),e\left(\frac{1}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 783 }(437, a) \) |
\(1\) | \(1\) | \(e\left(\frac{79}{252}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{127}{252}\right)\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{47}{252}\right)\) | \(e\left(\frac{16}{63}\right)\) |
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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)
