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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(287, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([20,21]))
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pari:[g,chi] = znchar(Mod(234,287))
| Modulus: | \(287\) | |
| Conductor: | \(287\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(120\) |
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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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\(\chi_{287}(12,\cdot)\)
\(\chi_{287}(17,\cdot)\)
\(\chi_{287}(19,\cdot)\)
\(\chi_{287}(24,\cdot)\)
\(\chi_{287}(26,\cdot)\)
\(\chi_{287}(47,\cdot)\)
\(\chi_{287}(52,\cdot)\)
\(\chi_{287}(54,\cdot)\)
\(\chi_{287}(75,\cdot)\)
\(\chi_{287}(89,\cdot)\)
\(\chi_{287}(94,\cdot)\)
\(\chi_{287}(101,\cdot)\)
\(\chi_{287}(108,\cdot)\)
\(\chi_{287}(110,\cdot)\)
\(\chi_{287}(117,\cdot)\)
\(\chi_{287}(129,\cdot)\)
\(\chi_{287}(136,\cdot)\)
\(\chi_{287}(138,\cdot)\)
\(\chi_{287}(145,\cdot)\)
\(\chi_{287}(152,\cdot)\)
\(\chi_{287}(157,\cdot)\)
\(\chi_{287}(171,\cdot)\)
\(\chi_{287}(192,\cdot)\)
\(\chi_{287}(194,\cdot)\)
\(\chi_{287}(199,\cdot)\)
\(\chi_{287}(220,\cdot)\)
\(\chi_{287}(222,\cdot)\)
\(\chi_{287}(227,\cdot)\)
\(\chi_{287}(229,\cdot)\)
\(\chi_{287}(234,\cdot)\)
...
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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 ]
\((206,211)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{7}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 287 }(234, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{67}{120}\right)\) |
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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)
