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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2912, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,21,16,4]))
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pari:[g,chi] = znchar(Mod(1005,2912))
| Modulus: | \(2912\) | |
| Conductor: | \(2912\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| 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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{2912}(205,\cdot)\)
\(\chi_{2912}(277,\cdot)\)
\(\chi_{2912}(933,\cdot)\)
\(\chi_{2912}(1005,\cdot)\)
\(\chi_{2912}(1661,\cdot)\)
\(\chi_{2912}(1733,\cdot)\)
\(\chi_{2912}(2389,\cdot)\)
\(\chi_{2912}(2461,\cdot)\)
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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 ]
\((2367,1093,1249,2017)\) → \((1,e\left(\frac{7}{8}\right),e\left(\frac{2}{3}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 2912 }(1005, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{7}{24}\right)\) | \(i\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{8}\right)\) |
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sage:chi.jacobi_sum(n)
