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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(473, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([84,50]))
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pari:[g,chi] = znchar(Mod(225,473))
| Modulus: | \(473\) | |
| Conductor: | \(473\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(105\) |
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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)
|
\(\chi_{473}(9,\cdot)\)
\(\chi_{473}(14,\cdot)\)
\(\chi_{473}(15,\cdot)\)
\(\chi_{473}(25,\cdot)\)
\(\chi_{473}(31,\cdot)\)
\(\chi_{473}(38,\cdot)\)
\(\chi_{473}(53,\cdot)\)
\(\chi_{473}(58,\cdot)\)
\(\chi_{473}(60,\cdot)\)
\(\chi_{473}(81,\cdot)\)
\(\chi_{473}(103,\cdot)\)
\(\chi_{473}(124,\cdot)\)
\(\chi_{473}(126,\cdot)\)
\(\chi_{473}(146,\cdot)\)
\(\chi_{473}(152,\cdot)\)
\(\chi_{473}(169,\cdot)\)
\(\chi_{473}(181,\cdot)\)
\(\chi_{473}(185,\cdot)\)
\(\chi_{473}(196,\cdot)\)
\(\chi_{473}(203,\cdot)\)
\(\chi_{473}(212,\cdot)\)
\(\chi_{473}(224,\cdot)\)
\(\chi_{473}(225,\cdot)\)
\(\chi_{473}(229,\cdot)\)
\(\chi_{473}(240,\cdot)\)
\(\chi_{473}(246,\cdot)\)
\(\chi_{473}(267,\cdot)\)
\(\chi_{473}(268,\cdot)\)
\(\chi_{473}(273,\cdot)\)
\(\chi_{473}(289,\cdot)\)
...
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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 ]
\((431,89)\) → \((e\left(\frac{2}{5}\right),e\left(\frac{5}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 473 }(225, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{92}{105}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) |
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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)
