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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([175,171]))
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pari:[g,chi] = znchar(Mod(33,497))
| 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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{497}(31,\cdot)\)
\(\chi_{497}(33,\cdot)\)
\(\chi_{497}(47,\cdot)\)
\(\chi_{497}(52,\cdot)\)
\(\chi_{497}(59,\cdot)\)
\(\chi_{497}(61,\cdot)\)
\(\chi_{497}(68,\cdot)\)
\(\chi_{497}(82,\cdot)\)
\(\chi_{497}(115,\cdot)\)
\(\chi_{497}(124,\cdot)\)
\(\chi_{497}(136,\cdot)\)
\(\chi_{497}(138,\cdot)\)
\(\chi_{497}(164,\cdot)\)
\(\chi_{497}(173,\cdot)\)
\(\chi_{497}(194,\cdot)\)
\(\chi_{497}(201,\cdot)\)
\(\chi_{497}(220,\cdot)\)
\(\chi_{497}(234,\cdot)\)
\(\chi_{497}(241,\cdot)\)
\(\chi_{497}(248,\cdot)\)
\(\chi_{497}(255,\cdot)\)
\(\chi_{497}(257,\cdot)\)
\(\chi_{497}(269,\cdot)\)
\(\chi_{497}(276,\cdot)\)
\(\chi_{497}(278,\cdot)\)
\(\chi_{497}(297,\cdot)\)
\(\chi_{497}(306,\cdot)\)
\(\chi_{497}(339,\cdot)\)
\(\chi_{497}(346,\cdot)\)
\(\chi_{497}(353,\cdot)\)
...
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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 ]
\((143,78)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{57}{70}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 497 }(33, a) \) |
\(1\) | \(1\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{109}{210}\right)\) | \(e\left(\frac{121}{210}\right)\) | \(e\left(\frac{23}{210}\right)\) |
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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)
