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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(896, base_ring=CyclotomicField(96))
M = H._module
chi = DirichletCharacter(H, M([48,93,16]))
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pari:[g,chi] = znchar(Mod(563,896))
| Modulus: | \(896\) | |
| Conductor: | \(896\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(96\) |
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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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\(\chi_{896}(3,\cdot)\)
\(\chi_{896}(19,\cdot)\)
\(\chi_{896}(59,\cdot)\)
\(\chi_{896}(75,\cdot)\)
\(\chi_{896}(115,\cdot)\)
\(\chi_{896}(131,\cdot)\)
\(\chi_{896}(171,\cdot)\)
\(\chi_{896}(187,\cdot)\)
\(\chi_{896}(227,\cdot)\)
\(\chi_{896}(243,\cdot)\)
\(\chi_{896}(283,\cdot)\)
\(\chi_{896}(299,\cdot)\)
\(\chi_{896}(339,\cdot)\)
\(\chi_{896}(355,\cdot)\)
\(\chi_{896}(395,\cdot)\)
\(\chi_{896}(411,\cdot)\)
\(\chi_{896}(451,\cdot)\)
\(\chi_{896}(467,\cdot)\)
\(\chi_{896}(507,\cdot)\)
\(\chi_{896}(523,\cdot)\)
\(\chi_{896}(563,\cdot)\)
\(\chi_{896}(579,\cdot)\)
\(\chi_{896}(619,\cdot)\)
\(\chi_{896}(635,\cdot)\)
\(\chi_{896}(675,\cdot)\)
\(\chi_{896}(691,\cdot)\)
\(\chi_{896}(731,\cdot)\)
\(\chi_{896}(747,\cdot)\)
\(\chi_{896}(787,\cdot)\)
\(\chi_{896}(803,\cdot)\)
...
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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 ]
\((127,645,129)\) → \((-1,e\left(\frac{31}{32}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 896 }(563, a) \) |
\(1\) | \(1\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) |
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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)
