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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(476, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,32,33]))
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pari:[g,chi] = znchar(Mod(347,476))
| Modulus: | \(476\) | |
| Conductor: | \(476\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(48\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{476}(11,\cdot)\)
\(\chi_{476}(23,\cdot)\)
\(\chi_{476}(39,\cdot)\)
\(\chi_{476}(79,\cdot)\)
\(\chi_{476}(95,\cdot)\)
\(\chi_{476}(107,\cdot)\)
\(\chi_{476}(163,\cdot)\)
\(\chi_{476}(207,\cdot)\)
\(\chi_{476}(235,\cdot)\)
\(\chi_{476}(275,\cdot)\)
\(\chi_{476}(303,\cdot)\)
\(\chi_{476}(347,\cdot)\)
\(\chi_{476}(403,\cdot)\)
\(\chi_{476}(415,\cdot)\)
\(\chi_{476}(431,\cdot)\)
\(\chi_{476}(471,\cdot)\)
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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 ]
\((239,409,309)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{11}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 476 }(347, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(-i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{9}{16}\right)\) |
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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)
