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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(711, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([52,48]))
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pari:[g,chi] = znchar(Mod(304,711))
| Modulus: | \(711\) | |
| Conductor: | \(711\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(39\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{711}(22,\cdot)\)
\(\chi_{711}(52,\cdot)\)
\(\chi_{711}(67,\cdot)\)
\(\chi_{711}(97,\cdot)\)
\(\chi_{711}(166,\cdot)\)
\(\chi_{711}(196,\cdot)\)
\(\chi_{711}(220,\cdot)\)
\(\chi_{711}(223,\cdot)\)
\(\chi_{711}(247,\cdot)\)
\(\chi_{711}(259,\cdot)\)
\(\chi_{711}(283,\cdot)\)
\(\chi_{711}(301,\cdot)\)
\(\chi_{711}(304,\cdot)\)
\(\chi_{711}(337,\cdot)\)
\(\chi_{711}(403,\cdot)\)
\(\chi_{711}(457,\cdot)\)
\(\chi_{711}(484,\cdot)\)
\(\chi_{711}(520,\cdot)\)
\(\chi_{711}(526,\cdot)\)
\(\chi_{711}(538,\cdot)\)
\(\chi_{711}(571,\cdot)\)
\(\chi_{711}(574,\cdot)\)
\(\chi_{711}(670,\cdot)\)
\(\chi_{711}(697,\cdot)\)
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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 ]
\((317,82)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{8}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 711 }(304, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) |
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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)
