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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([112,65]))
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pari:[g,chi] = znchar(Mod(14,319))
| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{319}(3,\cdot)\)
\(\chi_{319}(14,\cdot)\)
\(\chi_{319}(15,\cdot)\)
\(\chi_{319}(26,\cdot)\)
\(\chi_{319}(27,\cdot)\)
\(\chi_{319}(31,\cdot)\)
\(\chi_{319}(37,\cdot)\)
\(\chi_{319}(47,\cdot)\)
\(\chi_{319}(48,\cdot)\)
\(\chi_{319}(60,\cdot)\)
\(\chi_{319}(69,\cdot)\)
\(\chi_{319}(97,\cdot)\)
\(\chi_{319}(102,\cdot)\)
\(\chi_{319}(108,\cdot)\)
\(\chi_{319}(113,\cdot)\)
\(\chi_{319}(114,\cdot)\)
\(\chi_{319}(119,\cdot)\)
\(\chi_{319}(124,\cdot)\)
\(\chi_{319}(126,\cdot)\)
\(\chi_{319}(130,\cdot)\)
\(\chi_{319}(135,\cdot)\)
\(\chi_{319}(137,\cdot)\)
\(\chi_{319}(147,\cdot)\)
\(\chi_{319}(148,\cdot)\)
\(\chi_{319}(159,\cdot)\)
\(\chi_{319}(163,\cdot)\)
\(\chi_{319}(185,\cdot)\)
\(\chi_{319}(192,\cdot)\)
\(\chi_{319}(201,\cdot)\)
\(\chi_{319}(213,\cdot)\)
...
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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 ]
\((233,89)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{13}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 319 }(14, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{140}\right)\) | \(e\left(\frac{101}{140}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{111}{140}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(i\) |
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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)
