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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1045, base_ring=CyclotomicField(10))
M = H._module
chi = DirichletCharacter(H, M([5,1,5]))
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pari:[g,chi] = znchar(Mod(189,1045))
| Modulus: | \(1045\) | |
| Conductor: | \(1045\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(10\) |
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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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\(\chi_{1045}(94,\cdot)\)
\(\chi_{1045}(189,\cdot)\)
\(\chi_{1045}(569,\cdot)\)
\(\chi_{1045}(854,\cdot)\)
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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 ]
\((837,761,496)\) → \((-1,e\left(\frac{1}{10}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 1045 }(189, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) |
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sage:chi.jacobi_sum(n)
