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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(845, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([39,1]))
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pari:[g,chi] = znchar(Mod(2,845))
| Modulus: | \(845\) | |
| Conductor: | \(845\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(156\) |
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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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\(\chi_{845}(2,\cdot)\)
\(\chi_{845}(32,\cdot)\)
\(\chi_{845}(33,\cdot)\)
\(\chi_{845}(63,\cdot)\)
\(\chi_{845}(67,\cdot)\)
\(\chi_{845}(97,\cdot)\)
\(\chi_{845}(98,\cdot)\)
\(\chi_{845}(128,\cdot)\)
\(\chi_{845}(132,\cdot)\)
\(\chi_{845}(162,\cdot)\)
\(\chi_{845}(163,\cdot)\)
\(\chi_{845}(193,\cdot)\)
\(\chi_{845}(197,\cdot)\)
\(\chi_{845}(227,\cdot)\)
\(\chi_{845}(228,\cdot)\)
\(\chi_{845}(262,\cdot)\)
\(\chi_{845}(292,\cdot)\)
\(\chi_{845}(293,\cdot)\)
\(\chi_{845}(323,\cdot)\)
\(\chi_{845}(327,\cdot)\)
\(\chi_{845}(358,\cdot)\)
\(\chi_{845}(388,\cdot)\)
\(\chi_{845}(392,\cdot)\)
\(\chi_{845}(422,\cdot)\)
\(\chi_{845}(423,\cdot)\)
\(\chi_{845}(453,\cdot)\)
\(\chi_{845}(457,\cdot)\)
\(\chi_{845}(487,\cdot)\)
\(\chi_{845}(518,\cdot)\)
\(\chi_{845}(522,\cdot)\)
...
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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 ]
\((677,171)\) → \((i,e\left(\frac{1}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 845 }(2, a) \) |
\(1\) | \(1\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{85}{156}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{125}{156}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{103}{156}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{5}{26}\right)\) |
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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)
