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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(101, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([12]))
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pari:[g,chi] = znchar(Mod(5,101))
| Modulus: | \(101\) | |
| Conductor: | \(101\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(25\) |
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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
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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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\(\chi_{101}(5,\cdot)\)
\(\chi_{101}(16,\cdot)\)
\(\chi_{101}(19,\cdot)\)
\(\chi_{101}(24,\cdot)\)
\(\chi_{101}(25,\cdot)\)
\(\chi_{101}(31,\cdot)\)
\(\chi_{101}(37,\cdot)\)
\(\chi_{101}(52,\cdot)\)
\(\chi_{101}(54,\cdot)\)
\(\chi_{101}(56,\cdot)\)
\(\chi_{101}(58,\cdot)\)
\(\chi_{101}(68,\cdot)\)
\(\chi_{101}(71,\cdot)\)
\(\chi_{101}(78,\cdot)\)
\(\chi_{101}(79,\cdot)\)
\(\chi_{101}(80,\cdot)\)
\(\chi_{101}(81,\cdot)\)
\(\chi_{101}(88,\cdot)\)
\(\chi_{101}(92,\cdot)\)
\(\chi_{101}(97,\cdot)\)
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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 ]
\(2\) → \(e\left(\frac{6}{25}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 101 }(5, a) \) |
\(1\) | \(1\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(1\) | \(e\left(\frac{3}{25}\right)\) |
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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)
