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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([40,69]))
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pari:[g,chi] = znchar(Mod(30,287))
| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{287}(11,\cdot)\)
\(\chi_{287}(30,\cdot)\)
\(\chi_{287}(53,\cdot)\)
\(\chi_{287}(58,\cdot)\)
\(\chi_{287}(60,\cdot)\)
\(\chi_{287}(65,\cdot)\)
\(\chi_{287}(67,\cdot)\)
\(\chi_{287}(88,\cdot)\)
\(\chi_{287}(93,\cdot)\)
\(\chi_{287}(95,\cdot)\)
\(\chi_{287}(116,\cdot)\)
\(\chi_{287}(130,\cdot)\)
\(\chi_{287}(135,\cdot)\)
\(\chi_{287}(142,\cdot)\)
\(\chi_{287}(149,\cdot)\)
\(\chi_{287}(151,\cdot)\)
\(\chi_{287}(158,\cdot)\)
\(\chi_{287}(170,\cdot)\)
\(\chi_{287}(177,\cdot)\)
\(\chi_{287}(179,\cdot)\)
\(\chi_{287}(186,\cdot)\)
\(\chi_{287}(193,\cdot)\)
\(\chi_{287}(198,\cdot)\)
\(\chi_{287}(212,\cdot)\)
\(\chi_{287}(233,\cdot)\)
\(\chi_{287}(235,\cdot)\)
\(\chi_{287}(240,\cdot)\)
\(\chi_{287}(261,\cdot)\)
\(\chi_{287}(263,\cdot)\)
\(\chi_{287}(268,\cdot)\)
...
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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 ]
\((206,211)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{23}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 287 }(30, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{23}{120}\right)\) |
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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)
