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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([33,88,96]))
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pari:[g,chi] = znchar(Mod(417,805))
| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{805}(2,\cdot)\)
\(\chi_{805}(18,\cdot)\)
\(\chi_{805}(32,\cdot)\)
\(\chi_{805}(58,\cdot)\)
\(\chi_{805}(72,\cdot)\)
\(\chi_{805}(123,\cdot)\)
\(\chi_{805}(128,\cdot)\)
\(\chi_{805}(142,\cdot)\)
\(\chi_{805}(163,\cdot)\)
\(\chi_{805}(177,\cdot)\)
\(\chi_{805}(193,\cdot)\)
\(\chi_{805}(233,\cdot)\)
\(\chi_{805}(242,\cdot)\)
\(\chi_{805}(282,\cdot)\)
\(\chi_{805}(303,\cdot)\)
\(\chi_{805}(312,\cdot)\)
\(\chi_{805}(317,\cdot)\)
\(\chi_{805}(338,\cdot)\)
\(\chi_{805}(347,\cdot)\)
\(\chi_{805}(403,\cdot)\)
\(\chi_{805}(417,\cdot)\)
\(\chi_{805}(422,\cdot)\)
\(\chi_{805}(443,\cdot)\)
\(\chi_{805}(473,\cdot)\)
\(\chi_{805}(478,\cdot)\)
\(\chi_{805}(487,\cdot)\)
\(\chi_{805}(492,\cdot)\)
\(\chi_{805}(508,\cdot)\)
\(\chi_{805}(522,\cdot)\)
\(\chi_{805}(578,\cdot)\)
...
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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 ]
\((162,346,281)\) → \((i,e\left(\frac{2}{3}\right),e\left(\frac{8}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 805 }(417, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{132}\right)\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{5}{33}\right)\) |
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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)
