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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([85,21]))
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pari:[g,chi] = znchar(Mod(203,837))
| Modulus: | \(837\) | |
| Conductor: | \(837\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(90\) |
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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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\(\chi_{837}(11,\cdot)\)
\(\chi_{837}(83,\cdot)\)
\(\chi_{837}(86,\cdot)\)
\(\chi_{837}(137,\cdot)\)
\(\chi_{837}(146,\cdot)\)
\(\chi_{837}(158,\cdot)\)
\(\chi_{837}(167,\cdot)\)
\(\chi_{837}(203,\cdot)\)
\(\chi_{837}(290,\cdot)\)
\(\chi_{837}(362,\cdot)\)
\(\chi_{837}(365,\cdot)\)
\(\chi_{837}(416,\cdot)\)
\(\chi_{837}(425,\cdot)\)
\(\chi_{837}(437,\cdot)\)
\(\chi_{837}(446,\cdot)\)
\(\chi_{837}(482,\cdot)\)
\(\chi_{837}(569,\cdot)\)
\(\chi_{837}(641,\cdot)\)
\(\chi_{837}(644,\cdot)\)
\(\chi_{837}(695,\cdot)\)
\(\chi_{837}(704,\cdot)\)
\(\chi_{837}(716,\cdot)\)
\(\chi_{837}(725,\cdot)\)
\(\chi_{837}(761,\cdot)\)
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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 ]
\((218,406)\) → \((e\left(\frac{17}{18}\right),e\left(\frac{7}{30}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 837 }(203, a) \) |
\(1\) | \(1\) | \(e\left(\frac{49}{90}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{17}{90}\right)\) | \(e\left(\frac{8}{45}\right)\) |
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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)
