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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(164, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,1]))
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pari:[g,chi] = znchar(Mod(47,164))
| Modulus: | \(164\) | |
| Conductor: | \(164\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(40\) |
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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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\(\chi_{164}(7,\cdot)\)
\(\chi_{164}(11,\cdot)\)
\(\chi_{164}(15,\cdot)\)
\(\chi_{164}(19,\cdot)\)
\(\chi_{164}(35,\cdot)\)
\(\chi_{164}(47,\cdot)\)
\(\chi_{164}(63,\cdot)\)
\(\chi_{164}(67,\cdot)\)
\(\chi_{164}(71,\cdot)\)
\(\chi_{164}(75,\cdot)\)
\(\chi_{164}(95,\cdot)\)
\(\chi_{164}(99,\cdot)\)
\(\chi_{164}(111,\cdot)\)
\(\chi_{164}(135,\cdot)\)
\(\chi_{164}(147,\cdot)\)
\(\chi_{164}(151,\cdot)\)
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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 ]
\((83,129)\) → \((-1,e\left(\frac{1}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 164 }(47, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(-i\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) |
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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)
