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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(449, base_ring=CyclotomicField(112))
M = H._module
chi = DirichletCharacter(H, M([61]))
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pari:[g,chi] = znchar(Mod(82,449))
| Modulus: | \(449\) | |
| Conductor: | \(449\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(112\) |
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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)
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\(\chi_{449}(4,\cdot)\)
\(\chi_{449}(7,\cdot)\)
\(\chi_{449}(20,\cdot)\)
\(\chi_{449}(23,\cdot)\)
\(\chi_{449}(39,\cdot)\)
\(\chi_{449}(44,\cdot)\)
\(\chi_{449}(51,\cdot)\)
\(\chi_{449}(64,\cdot)\)
\(\chi_{449}(72,\cdot)\)
\(\chi_{449}(81,\cdot)\)
\(\chi_{449}(82,\cdot)\)
\(\chi_{449}(89,\cdot)\)
\(\chi_{449}(106,\cdot)\)
\(\chi_{449}(111,\cdot)\)
\(\chi_{449}(112,\cdot)\)
\(\chi_{449}(115,\cdot)\)
\(\chi_{449}(126,\cdot)\)
\(\chi_{449}(129,\cdot)\)
\(\chi_{449}(175,\cdot)\)
\(\chi_{449}(181,\cdot)\)
\(\chi_{449}(194,\cdot)\)
\(\chi_{449}(195,\cdot)\)
\(\chi_{449}(196,\cdot)\)
\(\chi_{449}(202,\cdot)\)
\(\chi_{449}(247,\cdot)\)
\(\chi_{449}(253,\cdot)\)
\(\chi_{449}(254,\cdot)\)
\(\chi_{449}(255,\cdot)\)
\(\chi_{449}(268,\cdot)\)
\(\chi_{449}(274,\cdot)\)
...
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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 ]
\(3\) → \(e\left(\frac{61}{112}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 449 }(82, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{61}{112}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{99}{112}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{14}\right)\) |
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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)
