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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(805, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([33,88,78]))
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pari:[g,chi] = znchar(Mod(67,805))
| Modulus: | \(805\) | |
| Conductor: | \(805\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(132\) |
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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
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{805}(37,\cdot)\)
\(\chi_{805}(53,\cdot)\)
\(\chi_{805}(67,\cdot)\)
\(\chi_{805}(88,\cdot)\)
\(\chi_{805}(102,\cdot)\)
\(\chi_{805}(107,\cdot)\)
\(\chi_{805}(158,\cdot)\)
\(\chi_{805}(172,\cdot)\)
\(\chi_{805}(198,\cdot)\)
\(\chi_{805}(212,\cdot)\)
\(\chi_{805}(228,\cdot)\)
\(\chi_{805}(247,\cdot)\)
\(\chi_{805}(263,\cdot)\)
\(\chi_{805}(268,\cdot)\)
\(\chi_{805}(333,\cdot)\)
\(\chi_{805}(352,\cdot)\)
\(\chi_{805}(373,\cdot)\)
\(\chi_{805}(382,\cdot)\)
\(\chi_{805}(387,\cdot)\)
\(\chi_{805}(408,\cdot)\)
\(\chi_{805}(452,\cdot)\)
\(\chi_{805}(457,\cdot)\)
\(\chi_{805}(513,\cdot)\)
\(\chi_{805}(527,\cdot)\)
\(\chi_{805}(543,\cdot)\)
\(\chi_{805}(548,\cdot)\)
\(\chi_{805}(557,\cdot)\)
\(\chi_{805}(562,\cdot)\)
\(\chi_{805}(592,\cdot)\)
\(\chi_{805}(613,\cdot)\)
...
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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 ]
\((162,346,281)\) → \((i,e\left(\frac{2}{3}\right),e\left(\frac{13}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 805 }(67, a) \) |
\(1\) | \(1\) | \(e\left(\frac{101}{132}\right)\) | \(e\left(\frac{115}{132}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{2}{33}\right)\) |
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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)
