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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(675, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([160,117]))
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pari:[g,chi] = znchar(Mod(142,675))
| Modulus: | \(675\) | |
| Conductor: | \(675\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(180\) |
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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)
|
\(\chi_{675}(13,\cdot)\)
\(\chi_{675}(22,\cdot)\)
\(\chi_{675}(52,\cdot)\)
\(\chi_{675}(58,\cdot)\)
\(\chi_{675}(67,\cdot)\)
\(\chi_{675}(88,\cdot)\)
\(\chi_{675}(97,\cdot)\)
\(\chi_{675}(103,\cdot)\)
\(\chi_{675}(112,\cdot)\)
\(\chi_{675}(133,\cdot)\)
\(\chi_{675}(142,\cdot)\)
\(\chi_{675}(148,\cdot)\)
\(\chi_{675}(178,\cdot)\)
\(\chi_{675}(187,\cdot)\)
\(\chi_{675}(202,\cdot)\)
\(\chi_{675}(223,\cdot)\)
\(\chi_{675}(238,\cdot)\)
\(\chi_{675}(247,\cdot)\)
\(\chi_{675}(277,\cdot)\)
\(\chi_{675}(283,\cdot)\)
\(\chi_{675}(292,\cdot)\)
\(\chi_{675}(313,\cdot)\)
\(\chi_{675}(322,\cdot)\)
\(\chi_{675}(328,\cdot)\)
\(\chi_{675}(337,\cdot)\)
\(\chi_{675}(358,\cdot)\)
\(\chi_{675}(367,\cdot)\)
\(\chi_{675}(373,\cdot)\)
\(\chi_{675}(403,\cdot)\)
\(\chi_{675}(412,\cdot)\)
...
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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 ]
\((326,352)\) → \((e\left(\frac{8}{9}\right),e\left(\frac{13}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 675 }(142, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{97}{180}\right)\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{83}{180}\right)\) | \(e\left(\frac{1}{90}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) |
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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)
