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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4284, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([6,8,4,3]))
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pari:[g,chi] = znchar(Mod(1339,4284))
| Modulus: | \(4284\) | |
| Conductor: | \(4284\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{4284}(319,\cdot)\)
\(\chi_{4284}(1075,\cdot)\)
\(\chi_{4284}(1339,\cdot)\)
\(\chi_{4284}(2095,\cdot)\)
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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 ]
\((2143,3809,1837,1261)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{1}{3}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4284 }(1339, a) \) |
\(-1\) | \(1\) | \(i\) | \(i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(i\) | \(-1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) |
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sage:chi.jacobi_sum(n)
