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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([12,3,4,12]))
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pari:[g,chi] = znchar(Mod(1403,2912))
| Modulus: | \(2912\) | |
| Conductor: | \(2912\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{2912}(467,\cdot)\)
\(\chi_{2912}(675,\cdot)\)
\(\chi_{2912}(1195,\cdot)\)
\(\chi_{2912}(1403,\cdot)\)
\(\chi_{2912}(1923,\cdot)\)
\(\chi_{2912}(2131,\cdot)\)
\(\chi_{2912}(2651,\cdot)\)
\(\chi_{2912}(2859,\cdot)\)
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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 ]
\((2367,1093,1249,2017)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{1}{6}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 2912 }(1403, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{8}\right)\) |
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sage:chi.jacobi_sum(n)
