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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1872, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([6,3,10,10]))
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pari:[g,chi] = znchar(Mod(491,1872))
| Modulus: | \(1872\) | |
| Conductor: | \(1872\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(12\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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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\(\chi_{1872}(491,\cdot)\)
\(\chi_{1872}(875,\cdot)\)
\(\chi_{1872}(1427,\cdot)\)
\(\chi_{1872}(1811,\cdot)\)
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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 ]
\((703,469,209,145)\) → \((-1,i,e\left(\frac{5}{6}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1872 }(491, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(-1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) |
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sage:chi.jacobi_sum(n)
