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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(763, base_ring=CyclotomicField(108))
M = H._module
chi = DirichletCharacter(H, M([18,25]))
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pari:[g,chi] = znchar(Mod(10,763))
| Modulus: | \(763\) | |
| Conductor: | \(763\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(108\) |
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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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\(\chi_{763}(10,\cdot)\)
\(\chi_{763}(24,\cdot)\)
\(\chi_{763}(47,\cdot)\)
\(\chi_{763}(52,\cdot)\)
\(\chi_{763}(115,\cdot)\)
\(\chi_{763}(122,\cdot)\)
\(\chi_{763}(159,\cdot)\)
\(\chi_{763}(178,\cdot)\)
\(\chi_{763}(229,\cdot)\)
\(\chi_{763}(236,\cdot)\)
\(\chi_{763}(257,\cdot)\)
\(\chi_{763}(269,\cdot)\)
\(\chi_{763}(276,\cdot)\)
\(\chi_{763}(285,\cdot)\)
\(\chi_{763}(367,\cdot)\)
\(\chi_{763}(369,\cdot)\)
\(\chi_{763}(397,\cdot)\)
\(\chi_{763}(418,\cdot)\)
\(\chi_{763}(423,\cdot)\)
\(\chi_{763}(425,\cdot)\)
\(\chi_{763}(430,\cdot)\)
\(\chi_{763}(493,\cdot)\)
\(\chi_{763}(495,\cdot)\)
\(\chi_{763}(521,\cdot)\)
\(\chi_{763}(535,\cdot)\)
\(\chi_{763}(598,\cdot)\)
\(\chi_{763}(607,\cdot)\)
\(\chi_{763}(640,\cdot)\)
\(\chi_{763}(668,\cdot)\)
\(\chi_{763}(684,\cdot)\)
...
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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 ]
\((437,442)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{25}{108}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 763 }(10, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{79}{108}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{103}{108}\right)\) | \(e\left(\frac{95}{108}\right)\) | \(e\left(\frac{7}{27}\right)\) |
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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)
