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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(22848, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,9,24,16,36]))
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pari:[g,chi] = znchar(Mod(18755,22848))
| Modulus: | \(22848\) | |
| Conductor: | \(22848\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(48\) |
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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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\(\chi_{22848}(1619,\cdot)\)
\(\chi_{22848}(2699,\cdot)\)
\(\chi_{22848}(4883,\cdot)\)
\(\chi_{22848}(5147,\cdot)\)
\(\chi_{22848}(7331,\cdot)\)
\(\chi_{22848}(8411,\cdot)\)
\(\chi_{22848}(10595,\cdot)\)
\(\chi_{22848}(10859,\cdot)\)
\(\chi_{22848}(13043,\cdot)\)
\(\chi_{22848}(14123,\cdot)\)
\(\chi_{22848}(16307,\cdot)\)
\(\chi_{22848}(16571,\cdot)\)
\(\chi_{22848}(18755,\cdot)\)
\(\chi_{22848}(19835,\cdot)\)
\(\chi_{22848}(22019,\cdot)\)
\(\chi_{22848}(22283,\cdot)\)
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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 ]
\((13567,18565,15233,3265,2689)\) → \((-1,e\left(\frac{3}{16}\right),-1,e\left(\frac{1}{3}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 22848 }(18755, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{3}{8}\right)\) |
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sage:chi.jacobi_sum(n)
