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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([196,27]))
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pari:[g,chi] = znchar(Mod(211,783))
| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{783}(31,\cdot)\)
\(\chi_{783}(40,\cdot)\)
\(\chi_{783}(43,\cdot)\)
\(\chi_{783}(61,\cdot)\)
\(\chi_{783}(76,\cdot)\)
\(\chi_{783}(79,\cdot)\)
\(\chi_{783}(85,\cdot)\)
\(\chi_{783}(97,\cdot)\)
\(\chi_{783}(106,\cdot)\)
\(\chi_{783}(124,\cdot)\)
\(\chi_{783}(130,\cdot)\)
\(\chi_{783}(142,\cdot)\)
\(\chi_{783}(148,\cdot)\)
\(\chi_{783}(160,\cdot)\)
\(\chi_{783}(166,\cdot)\)
\(\chi_{783}(184,\cdot)\)
\(\chi_{783}(193,\cdot)\)
\(\chi_{783}(205,\cdot)\)
\(\chi_{783}(211,\cdot)\)
\(\chi_{783}(214,\cdot)\)
\(\chi_{783}(229,\cdot)\)
\(\chi_{783}(247,\cdot)\)
\(\chi_{783}(250,\cdot)\)
\(\chi_{783}(259,\cdot)\)
\(\chi_{783}(292,\cdot)\)
\(\chi_{783}(301,\cdot)\)
\(\chi_{783}(304,\cdot)\)
\(\chi_{783}(322,\cdot)\)
\(\chi_{783}(337,\cdot)\)
\(\chi_{783}(340,\cdot)\)
...
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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 ]
\((407,379)\) → \((e\left(\frac{7}{9}\right),e\left(\frac{3}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 783 }(211, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{223}{252}\right)\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{199}{252}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{155}{252}\right)\) | \(e\left(\frac{34}{63}\right)\) |
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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)
