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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(763, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([45,29]))
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pari:[g,chi] = znchar(Mod(12,763))
| Modulus: | \(763\) | |
| Conductor: | \(763\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(54\) |
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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
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| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{763}(12,\cdot)\)
\(\chi_{763}(31,\cdot)\)
\(\chi_{763}(61,\cdot)\)
\(\chi_{763}(87,\cdot)\)
\(\chi_{763}(94,\cdot)\)
\(\chi_{763}(138,\cdot)\)
\(\chi_{763}(215,\cdot)\)
\(\chi_{763}(318,\cdot)\)
\(\chi_{763}(320,\cdot)\)
\(\chi_{763}(355,\cdot)\)
\(\chi_{763}(472,\cdot)\)
\(\chi_{763}(565,\cdot)\)
\(\chi_{763}(605,\cdot)\)
\(\chi_{763}(619,\cdot)\)
\(\chi_{763}(628,\cdot)\)
\(\chi_{763}(633,\cdot)\)
\(\chi_{763}(649,\cdot)\)
\(\chi_{763}(738,\cdot)\)
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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 ]
\((437,442)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{29}{54}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 763 }(12, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{17}{54}\right)\) |
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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)
