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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(837, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([80,24]))
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pari:[g,chi] = znchar(Mod(547,837))
| Modulus: | \(837\) | |
| Conductor: | \(837\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(45\) |
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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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\(\chi_{837}(76,\cdot)\)
\(\chi_{837}(112,\cdot)\)
\(\chi_{837}(121,\cdot)\)
\(\chi_{837}(133,\cdot)\)
\(\chi_{837}(142,\cdot)\)
\(\chi_{837}(193,\cdot)\)
\(\chi_{837}(196,\cdot)\)
\(\chi_{837}(268,\cdot)\)
\(\chi_{837}(355,\cdot)\)
\(\chi_{837}(391,\cdot)\)
\(\chi_{837}(400,\cdot)\)
\(\chi_{837}(412,\cdot)\)
\(\chi_{837}(421,\cdot)\)
\(\chi_{837}(472,\cdot)\)
\(\chi_{837}(475,\cdot)\)
\(\chi_{837}(547,\cdot)\)
\(\chi_{837}(634,\cdot)\)
\(\chi_{837}(670,\cdot)\)
\(\chi_{837}(679,\cdot)\)
\(\chi_{837}(691,\cdot)\)
\(\chi_{837}(700,\cdot)\)
\(\chi_{837}(751,\cdot)\)
\(\chi_{837}(754,\cdot)\)
\(\chi_{837}(826,\cdot)\)
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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 ]
\((218,406)\) → \((e\left(\frac{8}{9}\right),e\left(\frac{4}{15}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 837 }(547, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) |
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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)
