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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([22,126]))
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pari:[g,chi] = znchar(Mod(220,621))
| Modulus: | \(621\) | |
| Conductor: | \(621\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(99\) |
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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)
|
\(\chi_{621}(4,\cdot)\)
\(\chi_{621}(13,\cdot)\)
\(\chi_{621}(16,\cdot)\)
\(\chi_{621}(25,\cdot)\)
\(\chi_{621}(31,\cdot)\)
\(\chi_{621}(49,\cdot)\)
\(\chi_{621}(52,\cdot)\)
\(\chi_{621}(58,\cdot)\)
\(\chi_{621}(85,\cdot)\)
\(\chi_{621}(94,\cdot)\)
\(\chi_{621}(121,\cdot)\)
\(\chi_{621}(124,\cdot)\)
\(\chi_{621}(133,\cdot)\)
\(\chi_{621}(142,\cdot)\)
\(\chi_{621}(151,\cdot)\)
\(\chi_{621}(169,\cdot)\)
\(\chi_{621}(187,\cdot)\)
\(\chi_{621}(193,\cdot)\)
\(\chi_{621}(196,\cdot)\)
\(\chi_{621}(202,\cdot)\)
\(\chi_{621}(211,\cdot)\)
\(\chi_{621}(220,\cdot)\)
\(\chi_{621}(223,\cdot)\)
\(\chi_{621}(232,\cdot)\)
\(\chi_{621}(238,\cdot)\)
\(\chi_{621}(256,\cdot)\)
\(\chi_{621}(259,\cdot)\)
\(\chi_{621}(265,\cdot)\)
\(\chi_{621}(292,\cdot)\)
\(\chi_{621}(301,\cdot)\)
...
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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 ]
\((461,28)\) → \((e\left(\frac{1}{9}\right),e\left(\frac{7}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 621 }(220, a) \) |
\(1\) | \(1\) | \(e\left(\frac{38}{99}\right)\) | \(e\left(\frac{76}{99}\right)\) | \(e\left(\frac{19}{99}\right)\) | \(e\left(\frac{86}{99}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{17}{99}\right)\) | \(e\left(\frac{79}{99}\right)\) | \(e\left(\frac{25}{99}\right)\) | \(e\left(\frac{53}{99}\right)\) |
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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)
