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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([33,22,39]))
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pari:[g,chi] = znchar(Mod(67,828))
| 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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\(\chi_{828}(7,\cdot)\)
\(\chi_{828}(43,\cdot)\)
\(\chi_{828}(67,\cdot)\)
\(\chi_{828}(79,\cdot)\)
\(\chi_{828}(103,\cdot)\)
\(\chi_{828}(175,\cdot)\)
\(\chi_{828}(247,\cdot)\)
\(\chi_{828}(283,\cdot)\)
\(\chi_{828}(295,\cdot)\)
\(\chi_{828}(319,\cdot)\)
\(\chi_{828}(355,\cdot)\)
\(\chi_{828}(475,\cdot)\)
\(\chi_{828}(511,\cdot)\)
\(\chi_{828}(571,\cdot)\)
\(\chi_{828}(619,\cdot)\)
\(\chi_{828}(655,\cdot)\)
\(\chi_{828}(727,\cdot)\)
\(\chi_{828}(751,\cdot)\)
\(\chi_{828}(787,\cdot)\)
\(\chi_{828}(799,\cdot)\)
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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 ]
\((415,461,649)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{13}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 828 }(67, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) |
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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)
