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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(161, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,6]))
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pari:[g,chi] = znchar(Mod(25,161))
| Modulus: | \(161\) | |
| Conductor: | \(161\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(33\) |
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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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\(\chi_{161}(2,\cdot)\)
\(\chi_{161}(4,\cdot)\)
\(\chi_{161}(9,\cdot)\)
\(\chi_{161}(16,\cdot)\)
\(\chi_{161}(18,\cdot)\)
\(\chi_{161}(25,\cdot)\)
\(\chi_{161}(32,\cdot)\)
\(\chi_{161}(39,\cdot)\)
\(\chi_{161}(58,\cdot)\)
\(\chi_{161}(72,\cdot)\)
\(\chi_{161}(81,\cdot)\)
\(\chi_{161}(95,\cdot)\)
\(\chi_{161}(100,\cdot)\)
\(\chi_{161}(121,\cdot)\)
\(\chi_{161}(123,\cdot)\)
\(\chi_{161}(128,\cdot)\)
\(\chi_{161}(142,\cdot)\)
\(\chi_{161}(144,\cdot)\)
\(\chi_{161}(151,\cdot)\)
\(\chi_{161}(156,\cdot)\)
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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 ]
\((24,120)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 161 }(25, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) |
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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)
