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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4641, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([6,2,2,3]))
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pari:[g,chi] = znchar(Mod(3566,4641))
| Modulus: | \(4641\) | |
| Conductor: | \(4641\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(12\) |
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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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\(\chi_{4641}(803,\cdot)\)
\(\chi_{4641}(1109,\cdot)\)
\(\chi_{4641}(3260,\cdot)\)
\(\chi_{4641}(3566,\cdot)\)
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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 ]
\((3095,3979,3928,547)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{1}{6}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(16\) | \(19\) | \(20\) | \(22\) |
| \( \chi_{ 4641 }(3566, a) \) |
\(1\) | \(1\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(-1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) |
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sage:chi.jacobi_sum(n)
