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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(516, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,41]))
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pari:[g,chi] = znchar(Mod(287,516))
| Modulus: | \(516\) | |
| Conductor: | \(516\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(42\) |
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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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\(\chi_{516}(71,\cdot)\)
\(\chi_{516}(119,\cdot)\)
\(\chi_{516}(155,\cdot)\)
\(\chi_{516}(191,\cdot)\)
\(\chi_{516}(227,\cdot)\)
\(\chi_{516}(263,\cdot)\)
\(\chi_{516}(287,\cdot)\)
\(\chi_{516}(335,\cdot)\)
\(\chi_{516}(347,\cdot)\)
\(\chi_{516}(407,\cdot)\)
\(\chi_{516}(491,\cdot)\)
\(\chi_{516}(503,\cdot)\)
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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 ]
\((259,173,433)\) → \((-1,-1,e\left(\frac{41}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 516 }(287, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) |
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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)
