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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(765, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([32,12,9]))
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pari:[g,chi] = znchar(Mod(367,765))
| Modulus: | \(765\) | |
| Conductor: | \(765\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(48\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{765}(7,\cdot)\)
\(\chi_{765}(88,\cdot)\)
\(\chi_{765}(112,\cdot)\)
\(\chi_{765}(133,\cdot)\)
\(\chi_{765}(142,\cdot)\)
\(\chi_{765}(148,\cdot)\)
\(\chi_{765}(232,\cdot)\)
\(\chi_{765}(328,\cdot)\)
\(\chi_{765}(367,\cdot)\)
\(\chi_{765}(403,\cdot)\)
\(\chi_{765}(517,\cdot)\)
\(\chi_{765}(583,\cdot)\)
\(\chi_{765}(598,\cdot)\)
\(\chi_{765}(643,\cdot)\)
\(\chi_{765}(652,\cdot)\)
\(\chi_{765}(742,\cdot)\)
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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 ]
\((596,307,496)\) → \((e\left(\frac{2}{3}\right),i,e\left(\frac{3}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(19\) | \(22\) |
| \( \chi_{ 765 }(367, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{25}{48}\right)\) |
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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)
