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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(483, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,11,42]))
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pari:[g,chi] = znchar(Mod(59,483))
| Modulus: | \(483\) | |
| Conductor: | \(483\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| 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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\(\chi_{483}(26,\cdot)\)
\(\chi_{483}(59,\cdot)\)
\(\chi_{483}(101,\cdot)\)
\(\chi_{483}(110,\cdot)\)
\(\chi_{483}(131,\cdot)\)
\(\chi_{483}(164,\cdot)\)
\(\chi_{483}(173,\cdot)\)
\(\chi_{483}(215,\cdot)\)
\(\chi_{483}(236,\cdot)\)
\(\chi_{483}(248,\cdot)\)
\(\chi_{483}(257,\cdot)\)
\(\chi_{483}(269,\cdot)\)
\(\chi_{483}(278,\cdot)\)
\(\chi_{483}(311,\cdot)\)
\(\chi_{483}(353,\cdot)\)
\(\chi_{483}(374,\cdot)\)
\(\chi_{483}(395,\cdot)\)
\(\chi_{483}(404,\cdot)\)
\(\chi_{483}(416,\cdot)\)
\(\chi_{483}(446,\cdot)\)
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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 ]
\((323,346,442)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{7}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 483 }(59, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) |
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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)
