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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2223, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([3,3,5]))
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pari:[g,chi] = znchar(Mod(1343,2223))
| Modulus: | \(2223\) | |
| Conductor: | \(2223\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(18\) |
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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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\(\chi_{2223}(725,\cdot)\)
\(\chi_{2223}(1226,\cdot)\)
\(\chi_{2223}(1343,\cdot)\)
\(\chi_{2223}(1427,\cdot)\)
\(\chi_{2223}(1895,\cdot)\)
\(\chi_{2223}(2162,\cdot)\)
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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 ]
\((1730,1198,1522)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{1}{6}\right),e\left(\frac{5}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 2223 }(1343, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) |
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sage:chi.jacobi_sum(n)
