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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1001, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([2,6,1]))
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pari:[g,chi] = znchar(Mod(626,1001))
| Modulus: | \(1001\) | |
| Conductor: | \(1001\) |
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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)
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1001}(626,\cdot)\)
\(\chi_{1001}(670,\cdot)\)
\(\chi_{1001}(747,\cdot)\)
\(\chi_{1001}(934,\cdot)\)
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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 ]
\((430,365,925)\) → \((e\left(\frac{1}{6}\right),-1,e\left(\frac{1}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(15\) |
| \( \chi_{ 1001 }(626, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(-i\) | \(1\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) |
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sage:chi.jacobi_sum(n)
