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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(931, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([30,112]))
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pari:[g,chi] = znchar(Mod(347,931))
| Modulus: | \(931\) | |
| Conductor: | \(931\) |
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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)
|
\(\chi_{931}(9,\cdot)\)
\(\chi_{931}(23,\cdot)\)
\(\chi_{931}(44,\cdot)\)
\(\chi_{931}(74,\cdot)\)
\(\chi_{931}(81,\cdot)\)
\(\chi_{931}(130,\cdot)\)
\(\chi_{931}(142,\cdot)\)
\(\chi_{931}(156,\cdot)\)
\(\chi_{931}(207,\cdot)\)
\(\chi_{931}(289,\cdot)\)
\(\chi_{931}(310,\cdot)\)
\(\chi_{931}(340,\cdot)\)
\(\chi_{931}(347,\cdot)\)
\(\chi_{931}(396,\cdot)\)
\(\chi_{931}(408,\cdot)\)
\(\chi_{931}(443,\cdot)\)
\(\chi_{931}(473,\cdot)\)
\(\chi_{931}(480,\cdot)\)
\(\chi_{931}(529,\cdot)\)
\(\chi_{931}(541,\cdot)\)
\(\chi_{931}(555,\cdot)\)
\(\chi_{931}(576,\cdot)\)
\(\chi_{931}(613,\cdot)\)
\(\chi_{931}(662,\cdot)\)
\(\chi_{931}(674,\cdot)\)
\(\chi_{931}(688,\cdot)\)
\(\chi_{931}(709,\cdot)\)
\(\chi_{931}(739,\cdot)\)
\(\chi_{931}(746,\cdot)\)
\(\chi_{931}(795,\cdot)\)
...
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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 ]
\((248,344)\) → \((e\left(\frac{5}{21}\right),e\left(\frac{8}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 931 }(347, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) |
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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)
