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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(319, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([42,115]))
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pari:[g,chi] = znchar(Mod(184,319))
| Modulus: | \(319\) | |
| Conductor: | \(319\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(140\) |
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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)
|
\(\chi_{319}(2,\cdot)\)
\(\chi_{319}(8,\cdot)\)
\(\chi_{319}(18,\cdot)\)
\(\chi_{319}(19,\cdot)\)
\(\chi_{319}(39,\cdot)\)
\(\chi_{319}(40,\cdot)\)
\(\chi_{319}(50,\cdot)\)
\(\chi_{319}(61,\cdot)\)
\(\chi_{319}(68,\cdot)\)
\(\chi_{319}(72,\cdot)\)
\(\chi_{319}(73,\cdot)\)
\(\chi_{319}(79,\cdot)\)
\(\chi_{319}(84,\cdot)\)
\(\chi_{319}(85,\cdot)\)
\(\chi_{319}(90,\cdot)\)
\(\chi_{319}(95,\cdot)\)
\(\chi_{319}(101,\cdot)\)
\(\chi_{319}(105,\cdot)\)
\(\chi_{319}(106,\cdot)\)
\(\chi_{319}(118,\cdot)\)
\(\chi_{319}(127,\cdot)\)
\(\chi_{319}(134,\cdot)\)
\(\chi_{319}(156,\cdot)\)
\(\chi_{319}(160,\cdot)\)
\(\chi_{319}(171,\cdot)\)
\(\chi_{319}(172,\cdot)\)
\(\chi_{319}(182,\cdot)\)
\(\chi_{319}(184,\cdot)\)
\(\chi_{319}(189,\cdot)\)
\(\chi_{319}(193,\cdot)\)
...
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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 ]
\((233,89)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{23}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 319 }(184, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{140}\right)\) | \(e\left(\frac{71}{140}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{51}{140}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(-i\) |
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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)
