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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([33,22,45]))
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pari:[g,chi] = znchar(Mod(65,483))
| 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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\(\chi_{483}(11,\cdot)\)
\(\chi_{483}(44,\cdot)\)
\(\chi_{483}(53,\cdot)\)
\(\chi_{483}(65,\cdot)\)
\(\chi_{483}(74,\cdot)\)
\(\chi_{483}(86,\cdot)\)
\(\chi_{483}(107,\cdot)\)
\(\chi_{483}(149,\cdot)\)
\(\chi_{483}(158,\cdot)\)
\(\chi_{483}(191,\cdot)\)
\(\chi_{483}(212,\cdot)\)
\(\chi_{483}(221,\cdot)\)
\(\chi_{483}(263,\cdot)\)
\(\chi_{483}(296,\cdot)\)
\(\chi_{483}(359,\cdot)\)
\(\chi_{483}(389,\cdot)\)
\(\chi_{483}(401,\cdot)\)
\(\chi_{483}(410,\cdot)\)
\(\chi_{483}(431,\cdot)\)
\(\chi_{483}(452,\cdot)\)
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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 ]
\((323,346,442)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{15}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 483 }(65, a) \) |
\(1\) | \(1\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) |
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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)
