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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(288, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,3,8]))
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pari:[g,chi] = znchar(Mod(283,288))
| Modulus: | \(288\) | |
| Conductor: | \(288\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(24\) |
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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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\(\chi_{288}(43,\cdot)\)
\(\chi_{288}(67,\cdot)\)
\(\chi_{288}(115,\cdot)\)
\(\chi_{288}(139,\cdot)\)
\(\chi_{288}(187,\cdot)\)
\(\chi_{288}(211,\cdot)\)
\(\chi_{288}(259,\cdot)\)
\(\chi_{288}(283,\cdot)\)
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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 ]
\((127,37,65)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 288 }(283, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(-1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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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)
