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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(259, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([12,1]))
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pari:[g,chi] = znchar(Mod(2,259))
| Modulus: | \(259\) | |
| Conductor: | \(259\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(36\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| Real: | no |
| Primitive: | yes |
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sage:chi.is_primitive()
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pari:#znconreyconductor(g,chi)==1
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| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{259}(2,\cdot)\)
\(\chi_{259}(18,\cdot)\)
\(\chi_{259}(32,\cdot)\)
\(\chi_{259}(72,\cdot)\)
\(\chi_{259}(116,\cdot)\)
\(\chi_{259}(128,\cdot)\)
\(\chi_{259}(130,\cdot)\)
\(\chi_{259}(163,\cdot)\)
\(\chi_{259}(170,\cdot)\)
\(\chi_{259}(172,\cdot)\)
\(\chi_{259}(205,\cdot)\)
\(\chi_{259}(235,\cdot)\)
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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 ]
\((38,113)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{1}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 259 }(2, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(-i\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{9}\right)\) |
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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)
