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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1539, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([32,48]))
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pari:[g,chi] = znchar(Mod(1183,1539))
| Modulus: | \(1539\) | |
| Conductor: | \(1539\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(27\) |
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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
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1539}(25,\cdot)\)
\(\chi_{1539}(112,\cdot)\)
\(\chi_{1539}(142,\cdot)\)
\(\chi_{1539}(157,\cdot)\)
\(\chi_{1539}(232,\cdot)\)
\(\chi_{1539}(472,\cdot)\)
\(\chi_{1539}(538,\cdot)\)
\(\chi_{1539}(625,\cdot)\)
\(\chi_{1539}(655,\cdot)\)
\(\chi_{1539}(670,\cdot)\)
\(\chi_{1539}(745,\cdot)\)
\(\chi_{1539}(985,\cdot)\)
\(\chi_{1539}(1051,\cdot)\)
\(\chi_{1539}(1138,\cdot)\)
\(\chi_{1539}(1168,\cdot)\)
\(\chi_{1539}(1183,\cdot)\)
\(\chi_{1539}(1258,\cdot)\)
\(\chi_{1539}(1498,\cdot)\)
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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 ]
\((1217,325)\) → \((e\left(\frac{16}{27}\right),e\left(\frac{8}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1539 }(1183, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) |
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sage:chi.jacobi_sum(n)
