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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(473, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([63,60]))
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pari:[g,chi] = znchar(Mod(193,473))
| Modulus: | \(473\) | |
| Conductor: | \(473\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(70\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{473}(35,\cdot)\)
\(\chi_{473}(41,\cdot)\)
\(\chi_{473}(84,\cdot)\)
\(\chi_{473}(90,\cdot)\)
\(\chi_{473}(107,\cdot)\)
\(\chi_{473}(127,\cdot)\)
\(\chi_{473}(140,\cdot)\)
\(\chi_{473}(145,\cdot)\)
\(\chi_{473}(150,\cdot)\)
\(\chi_{473}(183,\cdot)\)
\(\chi_{473}(193,\cdot)\)
\(\chi_{473}(226,\cdot)\)
\(\chi_{473}(250,\cdot)\)
\(\chi_{473}(293,\cdot)\)
\(\chi_{473}(299,\cdot)\)
\(\chi_{473}(305,\cdot)\)
\(\chi_{473}(336,\cdot)\)
\(\chi_{473}(348,\cdot)\)
\(\chi_{473}(360,\cdot)\)
\(\chi_{473}(365,\cdot)\)
\(\chi_{473}(391,\cdot)\)
\(\chi_{473}(398,\cdot)\)
\(\chi_{473}(403,\cdot)\)
\(\chi_{473}(446,\cdot)\)
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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 ]
\((431,89)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{6}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 473 }(193, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) |
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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)
