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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(403, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([35,14]))
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pari:[g,chi] = znchar(Mod(141,403))
| Modulus: | \(403\) | |
| Conductor: | \(403\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(60\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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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\(\chi_{403}(24,\cdot)\)
\(\chi_{403}(84,\cdot)\)
\(\chi_{403}(137,\cdot)\)
\(\chi_{403}(141,\cdot)\)
\(\chi_{403}(158,\cdot)\)
\(\chi_{403}(167,\cdot)\)
\(\chi_{403}(176,\cdot)\)
\(\chi_{403}(210,\cdot)\)
\(\chi_{403}(228,\cdot)\)
\(\chi_{403}(292,\cdot)\)
\(\chi_{403}(327,\cdot)\)
\(\chi_{403}(331,\cdot)\)
\(\chi_{403}(332,\cdot)\)
\(\chi_{403}(353,\cdot)\)
\(\chi_{403}(375,\cdot)\)
\(\chi_{403}(383,\cdot)\)
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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 ]
\((249,313)\) → \((e\left(\frac{7}{12}\right),e\left(\frac{7}{30}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 403 }(141, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(-i\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) |
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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)
