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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(403, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([5,8]))
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pari:[g,chi] = znchar(Mod(8,403))
| Modulus: | \(403\) | |
| Conductor: | \(403\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(20\) |
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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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\(\chi_{403}(8,\cdot)\)
\(\chi_{403}(47,\cdot)\)
\(\chi_{403}(70,\cdot)\)
\(\chi_{403}(109,\cdot)\)
\(\chi_{403}(190,\cdot)\)
\(\chi_{403}(252,\cdot)\)
\(\chi_{403}(281,\cdot)\)
\(\chi_{403}(343,\cdot)\)
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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 ]
\((249,313)\) → \((i,e\left(\frac{2}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 403 }(8, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(i\) | \(i\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) |
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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)
