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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(805, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,55,27]))
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pari:[g,chi] = znchar(Mod(264,805))
| Modulus: | \(805\) | |
| Conductor: | \(805\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(66\) |
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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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\(\chi_{805}(19,\cdot)\)
\(\chi_{805}(89,\cdot)\)
\(\chi_{805}(129,\cdot)\)
\(\chi_{805}(159,\cdot)\)
\(\chi_{805}(194,\cdot)\)
\(\chi_{805}(199,\cdot)\)
\(\chi_{805}(264,\cdot)\)
\(\chi_{805}(304,\cdot)\)
\(\chi_{805}(339,\cdot)\)
\(\chi_{805}(444,\cdot)\)
\(\chi_{805}(474,\cdot)\)
\(\chi_{805}(479,\cdot)\)
\(\chi_{805}(544,\cdot)\)
\(\chi_{805}(549,\cdot)\)
\(\chi_{805}(619,\cdot)\)
\(\chi_{805}(649,\cdot)\)
\(\chi_{805}(654,\cdot)\)
\(\chi_{805}(684,\cdot)\)
\(\chi_{805}(724,\cdot)\)
\(\chi_{805}(789,\cdot)\)
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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 ]
\((162,346,281)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{9}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 805 }(264, a) \) |
\(1\) | \(1\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{31}{33}\right)\) |
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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)
