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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3751, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([17,11]))
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pari:[g,chi] = znchar(Mod(681,3751))
| Modulus: | \(3751\) | |
| Conductor: | \(3751\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(22\) |
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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
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{3751}(340,\cdot)\)
\(\chi_{3751}(681,\cdot)\)
\(\chi_{3751}(1022,\cdot)\)
\(\chi_{3751}(1363,\cdot)\)
\(\chi_{3751}(1704,\cdot)\)
\(\chi_{3751}(2045,\cdot)\)
\(\chi_{3751}(2386,\cdot)\)
\(\chi_{3751}(2727,\cdot)\)
\(\chi_{3751}(3068,\cdot)\)
\(\chi_{3751}(3409,\cdot)\)
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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 ]
\((2543,2421)\) → \((e\left(\frac{17}{22}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 3751 }(681, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{22}\right)\) | \(-1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) |
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sage:chi.jacobi_sum(n)
