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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(644, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,55,36]))
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pari:[g,chi] = znchar(Mod(271,644))
| Modulus: | \(644\) | |
| Conductor: | \(644\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(66\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{644}(3,\cdot)\)
\(\chi_{644}(31,\cdot)\)
\(\chi_{644}(59,\cdot)\)
\(\chi_{644}(75,\cdot)\)
\(\chi_{644}(87,\cdot)\)
\(\chi_{644}(131,\cdot)\)
\(\chi_{644}(187,\cdot)\)
\(\chi_{644}(215,\cdot)\)
\(\chi_{644}(243,\cdot)\)
\(\chi_{644}(255,\cdot)\)
\(\chi_{644}(271,\cdot)\)
\(\chi_{644}(311,\cdot)\)
\(\chi_{644}(395,\cdot)\)
\(\chi_{644}(423,\cdot)\)
\(\chi_{644}(439,\cdot)\)
\(\chi_{644}(495,\cdot)\)
\(\chi_{644}(535,\cdot)\)
\(\chi_{644}(579,\cdot)\)
\(\chi_{644}(591,\cdot)\)
\(\chi_{644}(607,\cdot)\)
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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 ]
\((323,185,281)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{6}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
| \( \chi_{ 644 }(271, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) |
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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)
