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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(497, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([25,12]))
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pari:[g,chi] = znchar(Mod(5,497))
| Modulus: | \(497\) | |
| Conductor: | \(497\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(30\) |
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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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\(\chi_{497}(5,\cdot)\)
\(\chi_{497}(54,\cdot)\)
\(\chi_{497}(96,\cdot)\)
\(\chi_{497}(199,\cdot)\)
\(\chi_{497}(341,\cdot)\)
\(\chi_{497}(360,\cdot)\)
\(\chi_{497}(409,\cdot)\)
\(\chi_{497}(451,\cdot)\)
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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 ]
\((143,78)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{2}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 497 }(5, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) |
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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)
