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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(539, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([40,42]))
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pari:[g,chi] = znchar(Mod(191,539))
| Modulus: | \(539\) | |
| Conductor: | \(539\) |
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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)
|
\(\chi_{539}(4,\cdot)\)
\(\chi_{539}(9,\cdot)\)
\(\chi_{539}(16,\cdot)\)
\(\chi_{539}(25,\cdot)\)
\(\chi_{539}(37,\cdot)\)
\(\chi_{539}(53,\cdot)\)
\(\chi_{539}(58,\cdot)\)
\(\chi_{539}(60,\cdot)\)
\(\chi_{539}(81,\cdot)\)
\(\chi_{539}(86,\cdot)\)
\(\chi_{539}(93,\cdot)\)
\(\chi_{539}(102,\cdot)\)
\(\chi_{539}(114,\cdot)\)
\(\chi_{539}(130,\cdot)\)
\(\chi_{539}(135,\cdot)\)
\(\chi_{539}(137,\cdot)\)
\(\chi_{539}(158,\cdot)\)
\(\chi_{539}(163,\cdot)\)
\(\chi_{539}(170,\cdot)\)
\(\chi_{539}(179,\cdot)\)
\(\chi_{539}(191,\cdot)\)
\(\chi_{539}(207,\cdot)\)
\(\chi_{539}(212,\cdot)\)
\(\chi_{539}(235,\cdot)\)
\(\chi_{539}(240,\cdot)\)
\(\chi_{539}(247,\cdot)\)
\(\chi_{539}(256,\cdot)\)
\(\chi_{539}(268,\cdot)\)
\(\chi_{539}(284,\cdot)\)
\(\chi_{539}(289,\cdot)\)
...
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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 ]
\((199,442)\) → \((e\left(\frac{4}{21}\right),e\left(\frac{1}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 539 }(191, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{17}{35}\right)\) |
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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)
