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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(783, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([28,36]))
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pari:[g,chi] = znchar(Mod(691,783))
| Modulus: | \(783\) | |
| Conductor: | \(783\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(63\) |
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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)
|
\(\chi_{783}(7,\cdot)\)
\(\chi_{783}(16,\cdot)\)
\(\chi_{783}(25,\cdot)\)
\(\chi_{783}(49,\cdot)\)
\(\chi_{783}(52,\cdot)\)
\(\chi_{783}(94,\cdot)\)
\(\chi_{783}(103,\cdot)\)
\(\chi_{783}(112,\cdot)\)
\(\chi_{783}(139,\cdot)\)
\(\chi_{783}(169,\cdot)\)
\(\chi_{783}(223,\cdot)\)
\(\chi_{783}(256,\cdot)\)
\(\chi_{783}(268,\cdot)\)
\(\chi_{783}(277,\cdot)\)
\(\chi_{783}(286,\cdot)\)
\(\chi_{783}(310,\cdot)\)
\(\chi_{783}(313,\cdot)\)
\(\chi_{783}(355,\cdot)\)
\(\chi_{783}(364,\cdot)\)
\(\chi_{783}(373,\cdot)\)
\(\chi_{783}(400,\cdot)\)
\(\chi_{783}(430,\cdot)\)
\(\chi_{783}(484,\cdot)\)
\(\chi_{783}(517,\cdot)\)
\(\chi_{783}(529,\cdot)\)
\(\chi_{783}(538,\cdot)\)
\(\chi_{783}(547,\cdot)\)
\(\chi_{783}(571,\cdot)\)
\(\chi_{783}(574,\cdot)\)
\(\chi_{783}(616,\cdot)\)
...
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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 ]
\((407,379)\) → \((e\left(\frac{2}{9}\right),e\left(\frac{2}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 783 }(691, a) \) |
\(1\) | \(1\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{2}{63}\right)\) |
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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)
