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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(432, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,14]))
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pari:[g,chi] = znchar(Mod(101,432))
| Modulus: | \(432\) | |
| Conductor: | \(432\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(36\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{432}(5,\cdot)\)
\(\chi_{432}(29,\cdot)\)
\(\chi_{432}(77,\cdot)\)
\(\chi_{432}(101,\cdot)\)
\(\chi_{432}(149,\cdot)\)
\(\chi_{432}(173,\cdot)\)
\(\chi_{432}(221,\cdot)\)
\(\chi_{432}(245,\cdot)\)
\(\chi_{432}(293,\cdot)\)
\(\chi_{432}(317,\cdot)\)
\(\chi_{432}(365,\cdot)\)
\(\chi_{432}(389,\cdot)\)
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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 ]
\((271,325,353)\) → \((1,i,e\left(\frac{7}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 432 }(101, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) |
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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)
