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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(169, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([5]))
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pari:[g,chi] = znchar(Mod(32,169))
| Modulus: | \(169\) | |
| Conductor: | \(169\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(156\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{169}(2,\cdot)\)
\(\chi_{169}(6,\cdot)\)
\(\chi_{169}(7,\cdot)\)
\(\chi_{169}(11,\cdot)\)
\(\chi_{169}(15,\cdot)\)
\(\chi_{169}(20,\cdot)\)
\(\chi_{169}(24,\cdot)\)
\(\chi_{169}(28,\cdot)\)
\(\chi_{169}(32,\cdot)\)
\(\chi_{169}(33,\cdot)\)
\(\chi_{169}(37,\cdot)\)
\(\chi_{169}(41,\cdot)\)
\(\chi_{169}(45,\cdot)\)
\(\chi_{169}(46,\cdot)\)
\(\chi_{169}(50,\cdot)\)
\(\chi_{169}(54,\cdot)\)
\(\chi_{169}(58,\cdot)\)
\(\chi_{169}(59,\cdot)\)
\(\chi_{169}(63,\cdot)\)
\(\chi_{169}(67,\cdot)\)
\(\chi_{169}(71,\cdot)\)
\(\chi_{169}(72,\cdot)\)
\(\chi_{169}(76,\cdot)\)
\(\chi_{169}(84,\cdot)\)
\(\chi_{169}(85,\cdot)\)
\(\chi_{169}(93,\cdot)\)
\(\chi_{169}(97,\cdot)\)
\(\chi_{169}(98,\cdot)\)
\(\chi_{169}(102,\cdot)\)
\(\chi_{169}(106,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{5}{156}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 169 }(32, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{156}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{1}{156}\right)\) | \(e\left(\frac{67}{156}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{47}{156}\right)\) |
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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)
