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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(828, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,11,54]))
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pari:[g,chi] = znchar(Mod(443,828))
| Modulus: | \(828\) | |
| Conductor: | \(828\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(66\) |
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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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\(\chi_{828}(59,\cdot)\)
\(\chi_{828}(95,\cdot)\)
\(\chi_{828}(119,\cdot)\)
\(\chi_{828}(131,\cdot)\)
\(\chi_{828}(167,\cdot)\)
\(\chi_{828}(239,\cdot)\)
\(\chi_{828}(311,\cdot)\)
\(\chi_{828}(335,\cdot)\)
\(\chi_{828}(347,\cdot)\)
\(\chi_{828}(371,\cdot)\)
\(\chi_{828}(407,\cdot)\)
\(\chi_{828}(443,\cdot)\)
\(\chi_{828}(455,\cdot)\)
\(\chi_{828}(491,\cdot)\)
\(\chi_{828}(515,\cdot)\)
\(\chi_{828}(587,\cdot)\)
\(\chi_{828}(623,\cdot)\)
\(\chi_{828}(671,\cdot)\)
\(\chi_{828}(731,\cdot)\)
\(\chi_{828}(767,\cdot)\)
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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 ]
\((415,461,649)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{9}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 828 }(443, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) |
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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)
