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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(621, base_ring=CyclotomicField(198))
M = H._module
chi = DirichletCharacter(H, M([121,9]))
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pari:[g,chi] = znchar(Mod(212,621))
| Modulus: | \(621\) | |
| Conductor: | \(621\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(198\) |
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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)
|
\(\chi_{621}(5,\cdot)\)
\(\chi_{621}(11,\cdot)\)
\(\chi_{621}(14,\cdot)\)
\(\chi_{621}(20,\cdot)\)
\(\chi_{621}(38,\cdot)\)
\(\chi_{621}(56,\cdot)\)
\(\chi_{621}(65,\cdot)\)
\(\chi_{621}(74,\cdot)\)
\(\chi_{621}(83,\cdot)\)
\(\chi_{621}(86,\cdot)\)
\(\chi_{621}(113,\cdot)\)
\(\chi_{621}(122,\cdot)\)
\(\chi_{621}(149,\cdot)\)
\(\chi_{621}(155,\cdot)\)
\(\chi_{621}(158,\cdot)\)
\(\chi_{621}(176,\cdot)\)
\(\chi_{621}(182,\cdot)\)
\(\chi_{621}(191,\cdot)\)
\(\chi_{621}(194,\cdot)\)
\(\chi_{621}(203,\cdot)\)
\(\chi_{621}(212,\cdot)\)
\(\chi_{621}(218,\cdot)\)
\(\chi_{621}(221,\cdot)\)
\(\chi_{621}(227,\cdot)\)
\(\chi_{621}(245,\cdot)\)
\(\chi_{621}(263,\cdot)\)
\(\chi_{621}(272,\cdot)\)
\(\chi_{621}(281,\cdot)\)
\(\chi_{621}(290,\cdot)\)
\(\chi_{621}(293,\cdot)\)
...
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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 ]
\((461,28)\) → \((e\left(\frac{11}{18}\right),e\left(\frac{1}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 621 }(212, a) \) |
\(1\) | \(1\) | \(e\left(\frac{139}{198}\right)\) | \(e\left(\frac{40}{99}\right)\) | \(e\left(\frac{10}{99}\right)\) | \(e\left(\frac{127}{198}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{35}{99}\right)\) | \(e\left(\frac{52}{99}\right)\) | \(e\left(\frac{34}{99}\right)\) | \(e\left(\frac{80}{99}\right)\) |
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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)
