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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(256, base_ring=CyclotomicField(64))
M = H._module
chi = DirichletCharacter(H, M([32,23]))
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pari:[g,chi] = znchar(Mod(19,256))
| Modulus: | \(256\) | |
| Conductor: | \(256\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(64\) |
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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)
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\(\chi_{256}(3,\cdot)\)
\(\chi_{256}(11,\cdot)\)
\(\chi_{256}(19,\cdot)\)
\(\chi_{256}(27,\cdot)\)
\(\chi_{256}(35,\cdot)\)
\(\chi_{256}(43,\cdot)\)
\(\chi_{256}(51,\cdot)\)
\(\chi_{256}(59,\cdot)\)
\(\chi_{256}(67,\cdot)\)
\(\chi_{256}(75,\cdot)\)
\(\chi_{256}(83,\cdot)\)
\(\chi_{256}(91,\cdot)\)
\(\chi_{256}(99,\cdot)\)
\(\chi_{256}(107,\cdot)\)
\(\chi_{256}(115,\cdot)\)
\(\chi_{256}(123,\cdot)\)
\(\chi_{256}(131,\cdot)\)
\(\chi_{256}(139,\cdot)\)
\(\chi_{256}(147,\cdot)\)
\(\chi_{256}(155,\cdot)\)
\(\chi_{256}(163,\cdot)\)
\(\chi_{256}(171,\cdot)\)
\(\chi_{256}(179,\cdot)\)
\(\chi_{256}(187,\cdot)\)
\(\chi_{256}(195,\cdot)\)
\(\chi_{256}(203,\cdot)\)
\(\chi_{256}(211,\cdot)\)
\(\chi_{256}(219,\cdot)\)
\(\chi_{256}(227,\cdot)\)
\(\chi_{256}(235,\cdot)\)
...
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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 ]
\((255,5)\) → \((-1,e\left(\frac{23}{64}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 256 }(19, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{11}{64}\right)\) |
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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)
