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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(644, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,22,27]))
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pari:[g,chi] = znchar(Mod(471,644))
| Modulus: | \(644\) | |
| Conductor: | \(644\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(66\) |
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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
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| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{644}(11,\cdot)\)
\(\chi_{644}(51,\cdot)\)
\(\chi_{644}(67,\cdot)\)
\(\chi_{644}(79,\cdot)\)
\(\chi_{644}(107,\cdot)\)
\(\chi_{644}(135,\cdot)\)
\(\chi_{644}(191,\cdot)\)
\(\chi_{644}(235,\cdot)\)
\(\chi_{644}(247,\cdot)\)
\(\chi_{644}(263,\cdot)\)
\(\chi_{644}(291,\cdot)\)
\(\chi_{644}(319,\cdot)\)
\(\chi_{644}(359,\cdot)\)
\(\chi_{644}(375,\cdot)\)
\(\chi_{644}(387,\cdot)\)
\(\chi_{644}(431,\cdot)\)
\(\chi_{644}(471,\cdot)\)
\(\chi_{644}(527,\cdot)\)
\(\chi_{644}(543,\cdot)\)
\(\chi_{644}(571,\cdot)\)
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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 ]
\((323,185,281)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{9}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
| \( \chi_{ 644 }(471, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) |
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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)
