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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(269, base_ring=CyclotomicField(134))
M = H._module
chi = DirichletCharacter(H, M([23]))
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pari:[g,chi] = znchar(Mod(170,269))
| Modulus: | \(269\) | |
| Conductor: | \(269\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(134\) |
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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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\(\chi_{269}(4,\cdot)\)
\(\chi_{269}(6,\cdot)\)
\(\chi_{269}(9,\cdot)\)
\(\chi_{269}(11,\cdot)\)
\(\chi_{269}(13,\cdot)\)
\(\chi_{269}(20,\cdot)\)
\(\chi_{269}(30,\cdot)\)
\(\chi_{269}(34,\cdot)\)
\(\chi_{269}(43,\cdot)\)
\(\chi_{269}(45,\cdot)\)
\(\chi_{269}(49,\cdot)\)
\(\chi_{269}(51,\cdot)\)
\(\chi_{269}(55,\cdot)\)
\(\chi_{269}(56,\cdot)\)
\(\chi_{269}(64,\cdot)\)
\(\chi_{269}(65,\cdot)\)
\(\chi_{269}(73,\cdot)\)
\(\chi_{269}(79,\cdot)\)
\(\chi_{269}(84,\cdot)\)
\(\chi_{269}(89,\cdot)\)
\(\chi_{269}(92,\cdot)\)
\(\chi_{269}(96,\cdot)\)
\(\chi_{269}(97,\cdot)\)
\(\chi_{269}(100,\cdot)\)
\(\chi_{269}(103,\cdot)\)
\(\chi_{269}(126,\cdot)\)
\(\chi_{269}(127,\cdot)\)
\(\chi_{269}(133,\cdot)\)
\(\chi_{269}(138,\cdot)\)
\(\chi_{269}(144,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{23}{134}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 269 }(170, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{134}\right)\) | \(e\left(\frac{95}{134}\right)\) | \(e\left(\frac{23}{67}\right)\) | \(e\left(\frac{47}{67}\right)\) | \(e\left(\frac{59}{67}\right)\) | \(e\left(\frac{35}{134}\right)\) | \(e\left(\frac{69}{134}\right)\) | \(e\left(\frac{28}{67}\right)\) | \(e\left(\frac{117}{134}\right)\) | \(e\left(\frac{32}{67}\right)\) |
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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)
