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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(445, base_ring=CyclotomicField(88))
M = H._module
chi = DirichletCharacter(H, M([22,65]))
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pari:[g,chi] = znchar(Mod(137,445))
| Modulus: | \(445\) | |
| Conductor: | \(445\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(88\) |
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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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\(\chi_{445}(7,\cdot)\)
\(\chi_{445}(13,\cdot)\)
\(\chi_{445}(38,\cdot)\)
\(\chi_{445}(58,\cdot)\)
\(\chi_{445}(63,\cdot)\)
\(\chi_{445}(82,\cdot)\)
\(\chi_{445}(92,\cdot)\)
\(\chi_{445}(108,\cdot)\)
\(\chi_{445}(112,\cdot)\)
\(\chi_{445}(117,\cdot)\)
\(\chi_{445}(118,\cdot)\)
\(\chi_{445}(122,\cdot)\)
\(\chi_{445}(132,\cdot)\)
\(\chi_{445}(137,\cdot)\)
\(\chi_{445}(143,\cdot)\)
\(\chi_{445}(148,\cdot)\)
\(\chi_{445}(163,\cdot)\)
\(\chi_{445}(172,\cdot)\)
\(\chi_{445}(192,\cdot)\)
\(\chi_{445}(193,\cdot)\)
\(\chi_{445}(202,\cdot)\)
\(\chi_{445}(208,\cdot)\)
\(\chi_{445}(213,\cdot)\)
\(\chi_{445}(238,\cdot)\)
\(\chi_{445}(248,\cdot)\)
\(\chi_{445}(293,\cdot)\)
\(\chi_{445}(298,\cdot)\)
\(\chi_{445}(318,\cdot)\)
\(\chi_{445}(332,\cdot)\)
\(\chi_{445}(342,\cdot)\)
...
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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 ]
\((357,181)\) → \((i,e\left(\frac{65}{88}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 445 }(137, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{43}{88}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{49}{88}\right)\) | \(e\left(\frac{7}{88}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{65}{88}\right)\) |
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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)
