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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([70,18]))
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pari:[g,chi] = znchar(Mod(2,497))
| Modulus: | \(497\) | |
| Conductor: | \(497\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(105\) |
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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)
|
\(\chi_{497}(2,\cdot)\)
\(\chi_{497}(4,\cdot)\)
\(\chi_{497}(9,\cdot)\)
\(\chi_{497}(16,\cdot)\)
\(\chi_{497}(18,\cdot)\)
\(\chi_{497}(58,\cdot)\)
\(\chi_{497}(60,\cdot)\)
\(\chi_{497}(74,\cdot)\)
\(\chi_{497}(79,\cdot)\)
\(\chi_{497}(81,\cdot)\)
\(\chi_{497}(86,\cdot)\)
\(\chi_{497}(95,\cdot)\)
\(\chi_{497}(100,\cdot)\)
\(\chi_{497}(107,\cdot)\)
\(\chi_{497}(109,\cdot)\)
\(\chi_{497}(114,\cdot)\)
\(\chi_{497}(121,\cdot)\)
\(\chi_{497}(135,\cdot)\)
\(\chi_{497}(144,\cdot)\)
\(\chi_{497}(151,\cdot)\)
\(\chi_{497}(158,\cdot)\)
\(\chi_{497}(191,\cdot)\)
\(\chi_{497}(200,\cdot)\)
\(\chi_{497}(219,\cdot)\)
\(\chi_{497}(221,\cdot)\)
\(\chi_{497}(228,\cdot)\)
\(\chi_{497}(240,\cdot)\)
\(\chi_{497}(242,\cdot)\)
\(\chi_{497}(249,\cdot)\)
\(\chi_{497}(256,\cdot)\)
...
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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 ]
\((143,78)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{3}{35}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 497 }(2, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{97}{105}\right)\) |
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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)
