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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(805, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,44,12]))
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pari:[g,chi] = znchar(Mod(4,805))
| Modulus: | \(805\) | |
| Conductor: | \(805\) |
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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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\(\chi_{805}(4,\cdot)\)
\(\chi_{805}(9,\cdot)\)
\(\chi_{805}(39,\cdot)\)
\(\chi_{805}(144,\cdot)\)
\(\chi_{805}(179,\cdot)\)
\(\chi_{805}(219,\cdot)\)
\(\chi_{805}(284,\cdot)\)
\(\chi_{805}(289,\cdot)\)
\(\chi_{805}(324,\cdot)\)
\(\chi_{805}(354,\cdot)\)
\(\chi_{805}(394,\cdot)\)
\(\chi_{805}(464,\cdot)\)
\(\chi_{805}(499,\cdot)\)
\(\chi_{805}(564,\cdot)\)
\(\chi_{805}(604,\cdot)\)
\(\chi_{805}(634,\cdot)\)
\(\chi_{805}(639,\cdot)\)
\(\chi_{805}(669,\cdot)\)
\(\chi_{805}(739,\cdot)\)
\(\chi_{805}(744,\cdot)\)
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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 ]
\((162,346,281)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{2}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 805 }(4, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{26}{33}\right)\) |
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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)
