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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([24,28]))
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pari:[g,chi] = znchar(Mod(681,931))
| 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)
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\(\chi_{931}(4,\cdot)\)
\(\chi_{931}(16,\cdot)\)
\(\chi_{931}(25,\cdot)\)
\(\chi_{931}(93,\cdot)\)
\(\chi_{931}(100,\cdot)\)
\(\chi_{931}(123,\cdot)\)
\(\chi_{931}(137,\cdot)\)
\(\chi_{931}(149,\cdot)\)
\(\chi_{931}(158,\cdot)\)
\(\chi_{931}(233,\cdot)\)
\(\chi_{931}(256,\cdot)\)
\(\chi_{931}(270,\cdot)\)
\(\chi_{931}(282,\cdot)\)
\(\chi_{931}(291,\cdot)\)
\(\chi_{931}(359,\cdot)\)
\(\chi_{931}(366,\cdot)\)
\(\chi_{931}(389,\cdot)\)
\(\chi_{931}(403,\cdot)\)
\(\chi_{931}(415,\cdot)\)
\(\chi_{931}(424,\cdot)\)
\(\chi_{931}(492,\cdot)\)
\(\chi_{931}(499,\cdot)\)
\(\chi_{931}(522,\cdot)\)
\(\chi_{931}(536,\cdot)\)
\(\chi_{931}(548,\cdot)\)
\(\chi_{931}(625,\cdot)\)
\(\chi_{931}(632,\cdot)\)
\(\chi_{931}(669,\cdot)\)
\(\chi_{931}(681,\cdot)\)
\(\chi_{931}(690,\cdot)\)
...
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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 ]
\((248,344)\) → \((e\left(\frac{4}{21}\right),e\left(\frac{2}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 931 }(681, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) |
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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)
