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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(497, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([35,144]))
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pari:[g,chi] = znchar(Mod(185,497))
| Modulus: | \(497\) | |
| Conductor: | \(497\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(210\) |
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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)
|
\(\chi_{497}(3,\cdot)\)
\(\chi_{497}(10,\cdot)\)
\(\chi_{497}(12,\cdot)\)
\(\chi_{497}(19,\cdot)\)
\(\chi_{497}(24,\cdot)\)
\(\chi_{497}(38,\cdot)\)
\(\chi_{497}(40,\cdot)\)
\(\chi_{497}(73,\cdot)\)
\(\chi_{497}(75,\cdot)\)
\(\chi_{497}(80,\cdot)\)
\(\chi_{497}(87,\cdot)\)
\(\chi_{497}(89,\cdot)\)
\(\chi_{497}(129,\cdot)\)
\(\chi_{497}(131,\cdot)\)
\(\chi_{497}(145,\cdot)\)
\(\chi_{497}(150,\cdot)\)
\(\chi_{497}(152,\cdot)\)
\(\chi_{497}(157,\cdot)\)
\(\chi_{497}(166,\cdot)\)
\(\chi_{497}(171,\cdot)\)
\(\chi_{497}(178,\cdot)\)
\(\chi_{497}(180,\cdot)\)
\(\chi_{497}(185,\cdot)\)
\(\chi_{497}(192,\cdot)\)
\(\chi_{497}(206,\cdot)\)
\(\chi_{497}(215,\cdot)\)
\(\chi_{497}(222,\cdot)\)
\(\chi_{497}(229,\cdot)\)
\(\chi_{497}(262,\cdot)\)
\(\chi_{497}(271,\cdot)\)
...
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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 ]
\((143,78)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{24}{35}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 497 }(185, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{101}{210}\right)\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{187}{210}\right)\) |
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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)
