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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(207, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([22,6]))
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pari:[g,chi] = znchar(Mod(94,207))
| Modulus: | \(207\) | |
| Conductor: | \(207\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(33\) |
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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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\(\chi_{207}(4,\cdot)\)
\(\chi_{207}(13,\cdot)\)
\(\chi_{207}(16,\cdot)\)
\(\chi_{207}(25,\cdot)\)
\(\chi_{207}(31,\cdot)\)
\(\chi_{207}(49,\cdot)\)
\(\chi_{207}(52,\cdot)\)
\(\chi_{207}(58,\cdot)\)
\(\chi_{207}(85,\cdot)\)
\(\chi_{207}(94,\cdot)\)
\(\chi_{207}(121,\cdot)\)
\(\chi_{207}(124,\cdot)\)
\(\chi_{207}(133,\cdot)\)
\(\chi_{207}(142,\cdot)\)
\(\chi_{207}(151,\cdot)\)
\(\chi_{207}(169,\cdot)\)
\(\chi_{207}(187,\cdot)\)
\(\chi_{207}(193,\cdot)\)
\(\chi_{207}(196,\cdot)\)
\(\chi_{207}(202,\cdot)\)
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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 ]
\((47,28)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{1}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 207 }(94, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) |
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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)
