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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([22,6]))
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pari:[g,chi] = znchar(Mod(177,847))
| Modulus: | \(847\) | |
| Conductor: | \(847\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(33\) |
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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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\(\chi_{847}(23,\cdot)\)
\(\chi_{847}(67,\cdot)\)
\(\chi_{847}(100,\cdot)\)
\(\chi_{847}(144,\cdot)\)
\(\chi_{847}(177,\cdot)\)
\(\chi_{847}(221,\cdot)\)
\(\chi_{847}(254,\cdot)\)
\(\chi_{847}(298,\cdot)\)
\(\chi_{847}(331,\cdot)\)
\(\chi_{847}(375,\cdot)\)
\(\chi_{847}(408,\cdot)\)
\(\chi_{847}(452,\cdot)\)
\(\chi_{847}(529,\cdot)\)
\(\chi_{847}(562,\cdot)\)
\(\chi_{847}(639,\cdot)\)
\(\chi_{847}(683,\cdot)\)
\(\chi_{847}(716,\cdot)\)
\(\chi_{847}(760,\cdot)\)
\(\chi_{847}(793,\cdot)\)
\(\chi_{847}(837,\cdot)\)
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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 ]
\((122,365)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{1}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 847 }(177, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) |
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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)
