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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(539, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([25,21]))
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pari:[g,chi] = znchar(Mod(164,539))
| Modulus: | \(539\) | |
| Conductor: | \(539\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(42\) |
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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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\(\chi_{539}(10,\cdot)\)
\(\chi_{539}(54,\cdot)\)
\(\chi_{539}(87,\cdot)\)
\(\chi_{539}(131,\cdot)\)
\(\chi_{539}(164,\cdot)\)
\(\chi_{539}(208,\cdot)\)
\(\chi_{539}(241,\cdot)\)
\(\chi_{539}(285,\cdot)\)
\(\chi_{539}(318,\cdot)\)
\(\chi_{539}(395,\cdot)\)
\(\chi_{539}(439,\cdot)\)
\(\chi_{539}(516,\cdot)\)
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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 ]
\((199,442)\) → \((e\left(\frac{25}{42}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 539 }(164, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{1}{7}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
