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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(460, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,11,4]))
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pari:[g,chi] = znchar(Mod(347,460))
| Modulus: | \(460\) | |
| Conductor: | \(460\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(44\) |
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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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\(\chi_{460}(3,\cdot)\)
\(\chi_{460}(27,\cdot)\)
\(\chi_{460}(87,\cdot)\)
\(\chi_{460}(123,\cdot)\)
\(\chi_{460}(127,\cdot)\)
\(\chi_{460}(147,\cdot)\)
\(\chi_{460}(163,\cdot)\)
\(\chi_{460}(167,\cdot)\)
\(\chi_{460}(187,\cdot)\)
\(\chi_{460}(223,\cdot)\)
\(\chi_{460}(243,\cdot)\)
\(\chi_{460}(303,\cdot)\)
\(\chi_{460}(307,\cdot)\)
\(\chi_{460}(347,\cdot)\)
\(\chi_{460}(363,\cdot)\)
\(\chi_{460}(403,\cdot)\)
\(\chi_{460}(407,\cdot)\)
\(\chi_{460}(423,\cdot)\)
\(\chi_{460}(427,\cdot)\)
\(\chi_{460}(443,\cdot)\)
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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 ]
\((231,277,281)\) → \((-1,i,e\left(\frac{1}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 460 }(347, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) |
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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)
