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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(415, base_ring=CyclotomicField(164))
M = H._module
chi = DirichletCharacter(H, M([41,94]))
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pari:[g,chi] = znchar(Mod(102,415))
| Modulus: | \(415\) | |
| Conductor: | \(415\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(164\) |
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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)
|
\(\chi_{415}(2,\cdot)\)
\(\chi_{415}(8,\cdot)\)
\(\chi_{415}(13,\cdot)\)
\(\chi_{415}(18,\cdot)\)
\(\chi_{415}(22,\cdot)\)
\(\chi_{415}(32,\cdot)\)
\(\chi_{415}(42,\cdot)\)
\(\chi_{415}(43,\cdot)\)
\(\chi_{415}(47,\cdot)\)
\(\chi_{415}(52,\cdot)\)
\(\chi_{415}(53,\cdot)\)
\(\chi_{415}(57,\cdot)\)
\(\chi_{415}(58,\cdot)\)
\(\chi_{415}(62,\cdot)\)
\(\chi_{415}(67,\cdot)\)
\(\chi_{415}(72,\cdot)\)
\(\chi_{415}(73,\cdot)\)
\(\chi_{415}(88,\cdot)\)
\(\chi_{415}(97,\cdot)\)
\(\chi_{415}(98,\cdot)\)
\(\chi_{415}(102,\cdot)\)
\(\chi_{415}(103,\cdot)\)
\(\chi_{415}(107,\cdot)\)
\(\chi_{415}(117,\cdot)\)
\(\chi_{415}(118,\cdot)\)
\(\chi_{415}(122,\cdot)\)
\(\chi_{415}(128,\cdot)\)
\(\chi_{415}(133,\cdot)\)
\(\chi_{415}(137,\cdot)\)
\(\chi_{415}(138,\cdot)\)
...
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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 ]
\((167,251)\) → \((i,e\left(\frac{47}{82}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 415 }(102, a) \) |
\(1\) | \(1\) | \(e\left(\frac{135}{164}\right)\) | \(e\left(\frac{3}{164}\right)\) | \(e\left(\frac{53}{82}\right)\) | \(e\left(\frac{69}{82}\right)\) | \(e\left(\frac{137}{164}\right)\) | \(e\left(\frac{77}{164}\right)\) | \(e\left(\frac{3}{82}\right)\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{109}{164}\right)\) | \(e\left(\frac{145}{164}\right)\) |
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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)
