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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(704, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,45,72]))
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pari:[g,chi] = znchar(Mod(347,704))
| Modulus: | \(704\) | |
| Conductor: | \(704\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(80\) |
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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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\(\chi_{704}(19,\cdot)\)
\(\chi_{704}(35,\cdot)\)
\(\chi_{704}(51,\cdot)\)
\(\chi_{704}(83,\cdot)\)
\(\chi_{704}(107,\cdot)\)
\(\chi_{704}(123,\cdot)\)
\(\chi_{704}(139,\cdot)\)
\(\chi_{704}(171,\cdot)\)
\(\chi_{704}(195,\cdot)\)
\(\chi_{704}(211,\cdot)\)
\(\chi_{704}(227,\cdot)\)
\(\chi_{704}(259,\cdot)\)
\(\chi_{704}(283,\cdot)\)
\(\chi_{704}(299,\cdot)\)
\(\chi_{704}(315,\cdot)\)
\(\chi_{704}(347,\cdot)\)
\(\chi_{704}(371,\cdot)\)
\(\chi_{704}(387,\cdot)\)
\(\chi_{704}(403,\cdot)\)
\(\chi_{704}(435,\cdot)\)
\(\chi_{704}(459,\cdot)\)
\(\chi_{704}(475,\cdot)\)
\(\chi_{704}(491,\cdot)\)
\(\chi_{704}(523,\cdot)\)
\(\chi_{704}(547,\cdot)\)
\(\chi_{704}(563,\cdot)\)
\(\chi_{704}(579,\cdot)\)
\(\chi_{704}(611,\cdot)\)
\(\chi_{704}(635,\cdot)\)
\(\chi_{704}(651,\cdot)\)
...
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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 ]
\((639,133,321)\) → \((-1,e\left(\frac{9}{16}\right),e\left(\frac{9}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 704 }(347, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) |
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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)
