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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(529, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([38]))
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pari:[g,chi] = znchar(Mod(300,529))
| Modulus: | \(529\) | |
| Conductor: | \(529\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(23\) |
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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
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| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{529}(24,\cdot)\)
\(\chi_{529}(47,\cdot)\)
\(\chi_{529}(70,\cdot)\)
\(\chi_{529}(93,\cdot)\)
\(\chi_{529}(116,\cdot)\)
\(\chi_{529}(139,\cdot)\)
\(\chi_{529}(162,\cdot)\)
\(\chi_{529}(185,\cdot)\)
\(\chi_{529}(208,\cdot)\)
\(\chi_{529}(231,\cdot)\)
\(\chi_{529}(254,\cdot)\)
\(\chi_{529}(277,\cdot)\)
\(\chi_{529}(300,\cdot)\)
\(\chi_{529}(323,\cdot)\)
\(\chi_{529}(346,\cdot)\)
\(\chi_{529}(369,\cdot)\)
\(\chi_{529}(392,\cdot)\)
\(\chi_{529}(415,\cdot)\)
\(\chi_{529}(438,\cdot)\)
\(\chi_{529}(461,\cdot)\)
\(\chi_{529}(484,\cdot)\)
\(\chi_{529}(507,\cdot)\)
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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 ]
\(5\) → \(e\left(\frac{19}{23}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 529 }(300, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) |
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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)
