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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(511, base_ring=CyclotomicField(72))
M = H._module
chi = DirichletCharacter(H, M([12,53]))
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pari:[g,chi] = znchar(Mod(199,511))
| Modulus: | \(511\) | |
| Conductor: | \(511\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(72\) |
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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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\(\chi_{511}(5,\cdot)\)
\(\chi_{511}(26,\cdot)\)
\(\chi_{511}(31,\cdot)\)
\(\chi_{511}(33,\cdot)\)
\(\chi_{511}(40,\cdot)\)
\(\chi_{511}(47,\cdot)\)
\(\chi_{511}(59,\cdot)\)
\(\chi_{511}(68,\cdot)\)
\(\chi_{511}(87,\cdot)\)
\(\chi_{511}(115,\cdot)\)
\(\chi_{511}(131,\cdot)\)
\(\chi_{511}(185,\cdot)\)
\(\chi_{511}(199,\cdot)\)
\(\chi_{511}(206,\cdot)\)
\(\chi_{511}(208,\cdot)\)
\(\chi_{511}(248,\cdot)\)
\(\chi_{511}(264,\cdot)\)
\(\chi_{511}(320,\cdot)\)
\(\chi_{511}(376,\cdot)\)
\(\chi_{511}(409,\cdot)\)
\(\chi_{511}(451,\cdot)\)
\(\chi_{511}(453,\cdot)\)
\(\chi_{511}(458,\cdot)\)
\(\chi_{511}(472,\cdot)\)
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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 ]
\((220,78)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{53}{72}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 511 }(199, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{1}{36}\right)\) |
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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)
