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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([70,177]))
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pari:[g,chi] = znchar(Mod(268,497))
| 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)
|
\(\chi_{497}(11,\cdot)\)
\(\chi_{497}(44,\cdot)\)
\(\chi_{497}(53,\cdot)\)
\(\chi_{497}(65,\cdot)\)
\(\chi_{497}(67,\cdot)\)
\(\chi_{497}(93,\cdot)\)
\(\chi_{497}(102,\cdot)\)
\(\chi_{497}(123,\cdot)\)
\(\chi_{497}(130,\cdot)\)
\(\chi_{497}(149,\cdot)\)
\(\chi_{497}(163,\cdot)\)
\(\chi_{497}(170,\cdot)\)
\(\chi_{497}(177,\cdot)\)
\(\chi_{497}(184,\cdot)\)
\(\chi_{497}(186,\cdot)\)
\(\chi_{497}(198,\cdot)\)
\(\chi_{497}(205,\cdot)\)
\(\chi_{497}(207,\cdot)\)
\(\chi_{497}(226,\cdot)\)
\(\chi_{497}(235,\cdot)\)
\(\chi_{497}(268,\cdot)\)
\(\chi_{497}(275,\cdot)\)
\(\chi_{497}(282,\cdot)\)
\(\chi_{497}(291,\cdot)\)
\(\chi_{497}(305,\cdot)\)
\(\chi_{497}(312,\cdot)\)
\(\chi_{497}(317,\cdot)\)
\(\chi_{497}(319,\cdot)\)
\(\chi_{497}(326,\cdot)\)
\(\chi_{497}(331,\cdot)\)
...
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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 ]
\((143,78)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{59}{70}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 497 }(268, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{97}{210}\right)\) | \(e\left(\frac{73}{105}\right)\) |
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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)
