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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1309, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([16,12,9]))
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pari:[g,chi] = znchar(Mod(32,1309))
| Modulus: | \(1309\) | |
| Conductor: | \(1309\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(24\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1309}(32,\cdot)\)
\(\chi_{1309}(219,\cdot)\)
\(\chi_{1309}(263,\cdot)\)
\(\chi_{1309}(417,\cdot)\)
\(\chi_{1309}(450,\cdot)\)
\(\chi_{1309}(604,\cdot)\)
\(\chi_{1309}(648,\cdot)\)
\(\chi_{1309}(835,\cdot)\)
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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 ]
\((1123,596,309)\) → \((e\left(\frac{2}{3}\right),-1,e\left(\frac{3}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(32, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(1\) |
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sage:chi.jacobi_sum(n)
