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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([40,78]))
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pari:[g,chi] = znchar(Mod(607,837))
| 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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\(\chi_{837}(7,\cdot)\)
\(\chi_{837}(40,\cdot)\)
\(\chi_{837}(49,\cdot)\)
\(\chi_{837}(103,\cdot)\)
\(\chi_{837}(169,\cdot)\)
\(\chi_{837}(175,\cdot)\)
\(\chi_{837}(205,\cdot)\)
\(\chi_{837}(214,\cdot)\)
\(\chi_{837}(286,\cdot)\)
\(\chi_{837}(319,\cdot)\)
\(\chi_{837}(328,\cdot)\)
\(\chi_{837}(382,\cdot)\)
\(\chi_{837}(448,\cdot)\)
\(\chi_{837}(454,\cdot)\)
\(\chi_{837}(484,\cdot)\)
\(\chi_{837}(493,\cdot)\)
\(\chi_{837}(565,\cdot)\)
\(\chi_{837}(598,\cdot)\)
\(\chi_{837}(607,\cdot)\)
\(\chi_{837}(661,\cdot)\)
\(\chi_{837}(727,\cdot)\)
\(\chi_{837}(733,\cdot)\)
\(\chi_{837}(763,\cdot)\)
\(\chi_{837}(772,\cdot)\)
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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 ]
\((218,406)\) → \((e\left(\frac{4}{9}\right),e\left(\frac{13}{15}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 837 }(607, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) |
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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)
