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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([70,72]))
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pari:[g,chi] = znchar(Mod(157,837))
| 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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\(\chi_{837}(4,\cdot)\)
\(\chi_{837}(16,\cdot)\)
\(\chi_{837}(70,\cdot)\)
\(\chi_{837}(97,\cdot)\)
\(\chi_{837}(157,\cdot)\)
\(\chi_{837}(202,\cdot)\)
\(\chi_{837}(250,\cdot)\)
\(\chi_{837}(256,\cdot)\)
\(\chi_{837}(283,\cdot)\)
\(\chi_{837}(295,\cdot)\)
\(\chi_{837}(349,\cdot)\)
\(\chi_{837}(376,\cdot)\)
\(\chi_{837}(436,\cdot)\)
\(\chi_{837}(481,\cdot)\)
\(\chi_{837}(529,\cdot)\)
\(\chi_{837}(535,\cdot)\)
\(\chi_{837}(562,\cdot)\)
\(\chi_{837}(574,\cdot)\)
\(\chi_{837}(628,\cdot)\)
\(\chi_{837}(655,\cdot)\)
\(\chi_{837}(715,\cdot)\)
\(\chi_{837}(760,\cdot)\)
\(\chi_{837}(808,\cdot)\)
\(\chi_{837}(814,\cdot)\)
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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 ]
\((218,406)\) → \((e\left(\frac{7}{9}\right),e\left(\frac{4}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 837 }(157, a) \) |
\(1\) | \(1\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) |
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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)
