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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([99,110,84]))
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pari:[g,chi] = znchar(Mod(243,805))
| 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)
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\(\chi_{805}(3,\cdot)\)
\(\chi_{805}(12,\cdot)\)
\(\chi_{805}(52,\cdot)\)
\(\chi_{805}(73,\cdot)\)
\(\chi_{805}(82,\cdot)\)
\(\chi_{805}(87,\cdot)\)
\(\chi_{805}(108,\cdot)\)
\(\chi_{805}(117,\cdot)\)
\(\chi_{805}(173,\cdot)\)
\(\chi_{805}(187,\cdot)\)
\(\chi_{805}(192,\cdot)\)
\(\chi_{805}(213,\cdot)\)
\(\chi_{805}(243,\cdot)\)
\(\chi_{805}(248,\cdot)\)
\(\chi_{805}(257,\cdot)\)
\(\chi_{805}(262,\cdot)\)
\(\chi_{805}(278,\cdot)\)
\(\chi_{805}(292,\cdot)\)
\(\chi_{805}(348,\cdot)\)
\(\chi_{805}(353,\cdot)\)
\(\chi_{805}(397,\cdot)\)
\(\chi_{805}(418,\cdot)\)
\(\chi_{805}(423,\cdot)\)
\(\chi_{805}(432,\cdot)\)
\(\chi_{805}(453,\cdot)\)
\(\chi_{805}(472,\cdot)\)
\(\chi_{805}(537,\cdot)\)
\(\chi_{805}(542,\cdot)\)
\(\chi_{805}(558,\cdot)\)
\(\chi_{805}(577,\cdot)\)
...
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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 ]
\((162,346,281)\) → \((-i,e\left(\frac{5}{6}\right),e\left(\frac{7}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 805 }(243, a) \) |
\(1\) | \(1\) | \(e\left(\frac{91}{132}\right)\) | \(e\left(\frac{35}{132}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{85}{132}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{25}{33}\right)\) |
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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)
