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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(340, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,4,9]))
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pari:[g,chi] = znchar(Mod(167,340))
| Modulus: | \(340\) | |
| Conductor: | \(340\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(16\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{340}(23,\cdot)\)
\(\chi_{340}(107,\cdot)\)
\(\chi_{340}(143,\cdot)\)
\(\chi_{340}(163,\cdot)\)
\(\chi_{340}(167,\cdot)\)
\(\chi_{340}(207,\cdot)\)
\(\chi_{340}(267,\cdot)\)
\(\chi_{340}(283,\cdot)\)
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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 ]
\((171,137,241)\) → \((-1,i,e\left(\frac{9}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 340 }(167, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) |
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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)
