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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(595, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([36,32,45]))
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pari:[g,chi] = znchar(Mod(193,595))
| Modulus: | \(595\) | |
| Conductor: | \(595\) |
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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)
|
| 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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\(\chi_{595}(23,\cdot)\)
\(\chi_{595}(37,\cdot)\)
\(\chi_{595}(58,\cdot)\)
\(\chi_{595}(107,\cdot)\)
\(\chi_{595}(163,\cdot)\)
\(\chi_{595}(193,\cdot)\)
\(\chi_{595}(198,\cdot)\)
\(\chi_{595}(207,\cdot)\)
\(\chi_{595}(228,\cdot)\)
\(\chi_{595}(277,\cdot)\)
\(\chi_{595}(333,\cdot)\)
\(\chi_{595}(352,\cdot)\)
\(\chi_{595}(368,\cdot)\)
\(\chi_{595}(422,\cdot)\)
\(\chi_{595}(522,\cdot)\)
\(\chi_{595}(592,\cdot)\)
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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 ]
\((477,171,71)\) → \((-i,e\left(\frac{2}{3}\right),e\left(\frac{15}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 595 }(193, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(1\) | \(e\left(\frac{5}{6}\right)\) |
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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)
