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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(623, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,24]))
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pari:[g,chi] = znchar(Mod(4,623))
| Modulus: | \(623\) | |
| Conductor: | \(623\) |
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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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\(\chi_{623}(2,\cdot)\)
\(\chi_{623}(4,\cdot)\)
\(\chi_{623}(16,\cdot)\)
\(\chi_{623}(32,\cdot)\)
\(\chi_{623}(39,\cdot)\)
\(\chi_{623}(67,\cdot)\)
\(\chi_{623}(93,\cdot)\)
\(\chi_{623}(121,\cdot)\)
\(\chi_{623}(128,\cdot)\)
\(\chi_{623}(156,\cdot)\)
\(\chi_{623}(186,\cdot)\)
\(\chi_{623}(242,\cdot)\)
\(\chi_{623}(256,\cdot)\)
\(\chi_{623}(275,\cdot)\)
\(\chi_{623}(312,\cdot)\)
\(\chi_{623}(331,\cdot)\)
\(\chi_{623}(345,\cdot)\)
\(\chi_{623}(401,\cdot)\)
\(\chi_{623}(536,\cdot)\)
\(\chi_{623}(550,\cdot)\)
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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 ]
\((535,92)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{4}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 623 }(4, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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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)
