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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(539, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([10,42]))
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pari:[g,chi] = znchar(Mod(141,539))
| Modulus: | \(539\) | |
| Conductor: | \(539\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(35\) |
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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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\(\chi_{539}(15,\cdot)\)
\(\chi_{539}(36,\cdot)\)
\(\chi_{539}(64,\cdot)\)
\(\chi_{539}(71,\cdot)\)
\(\chi_{539}(92,\cdot)\)
\(\chi_{539}(113,\cdot)\)
\(\chi_{539}(141,\cdot)\)
\(\chi_{539}(169,\cdot)\)
\(\chi_{539}(190,\cdot)\)
\(\chi_{539}(218,\cdot)\)
\(\chi_{539}(225,\cdot)\)
\(\chi_{539}(267,\cdot)\)
\(\chi_{539}(302,\cdot)\)
\(\chi_{539}(323,\cdot)\)
\(\chi_{539}(372,\cdot)\)
\(\chi_{539}(379,\cdot)\)
\(\chi_{539}(400,\cdot)\)
\(\chi_{539}(421,\cdot)\)
\(\chi_{539}(449,\cdot)\)
\(\chi_{539}(456,\cdot)\)
\(\chi_{539}(477,\cdot)\)
\(\chi_{539}(498,\cdot)\)
\(\chi_{539}(526,\cdot)\)
\(\chi_{539}(533,\cdot)\)
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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 ]
\((199,442)\) → \((e\left(\frac{1}{7}\right),e\left(\frac{3}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 539 }(141, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{11}{35}\right)\) |
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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)
