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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6384, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,18,30,4]))
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pari:[g,chi] = znchar(Mod(2189,6384))
| Modulus: | \(6384\) | |
| Conductor: | \(6384\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(36\) |
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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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\(\chi_{6384}(605,\cdot)\)
\(\chi_{6384}(845,\cdot)\)
\(\chi_{6384}(1013,\cdot)\)
\(\chi_{6384}(1613,\cdot)\)
\(\chi_{6384}(2189,\cdot)\)
\(\chi_{6384}(2285,\cdot)\)
\(\chi_{6384}(3797,\cdot)\)
\(\chi_{6384}(4037,\cdot)\)
\(\chi_{6384}(4205,\cdot)\)
\(\chi_{6384}(4805,\cdot)\)
\(\chi_{6384}(5381,\cdot)\)
\(\chi_{6384}(5477,\cdot)\)
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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 ]
\((799,4789,2129,913,1009)\) → \((1,-i,-1,e\left(\frac{5}{6}\right),e\left(\frac{1}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 6384 }(2189, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(-1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{18}\right)\) |
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sage:chi.jacobi_sum(n)
