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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(783, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([70,45]))
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pari:[g,chi] = znchar(Mod(241,783))
| Modulus: | \(783\) | |
| Conductor: | \(783\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(126\) |
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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)
|
\(\chi_{783}(4,\cdot)\)
\(\chi_{783}(13,\cdot)\)
\(\chi_{783}(22,\cdot)\)
\(\chi_{783}(34,\cdot)\)
\(\chi_{783}(67,\cdot)\)
\(\chi_{783}(121,\cdot)\)
\(\chi_{783}(151,\cdot)\)
\(\chi_{783}(178,\cdot)\)
\(\chi_{783}(187,\cdot)\)
\(\chi_{783}(196,\cdot)\)
\(\chi_{783}(238,\cdot)\)
\(\chi_{783}(241,\cdot)\)
\(\chi_{783}(265,\cdot)\)
\(\chi_{783}(274,\cdot)\)
\(\chi_{783}(283,\cdot)\)
\(\chi_{783}(295,\cdot)\)
\(\chi_{783}(328,\cdot)\)
\(\chi_{783}(382,\cdot)\)
\(\chi_{783}(412,\cdot)\)
\(\chi_{783}(439,\cdot)\)
\(\chi_{783}(448,\cdot)\)
\(\chi_{783}(457,\cdot)\)
\(\chi_{783}(499,\cdot)\)
\(\chi_{783}(502,\cdot)\)
\(\chi_{783}(526,\cdot)\)
\(\chi_{783}(535,\cdot)\)
\(\chi_{783}(544,\cdot)\)
\(\chi_{783}(556,\cdot)\)
\(\chi_{783}(589,\cdot)\)
\(\chi_{783}(643,\cdot)\)
...
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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 ]
\((407,379)\) → \((e\left(\frac{5}{9}\right),e\left(\frac{5}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 783 }(241, a) \) |
\(1\) | \(1\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{11}{126}\right)\) | \(e\left(\frac{41}{63}\right)\) |
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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)
