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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4896, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,21,8,9]))
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pari:[g,chi] = znchar(Mod(4333,4896))
| Modulus: | \(4896\) | |
| Conductor: | \(4896\) |
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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)
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| 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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\(\chi_{4896}(229,\cdot)\)
\(\chi_{4896}(733,\cdot)\)
\(\chi_{4896}(1069,\cdot)\)
\(\chi_{4896}(1861,\cdot)\)
\(\chi_{4896}(2365,\cdot)\)
\(\chi_{4896}(2389,\cdot)\)
\(\chi_{4896}(4021,\cdot)\)
\(\chi_{4896}(4333,\cdot)\)
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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 ]
\((2143,613,3809,4321)\) → \((1,e\left(\frac{7}{8}\right),e\left(\frac{1}{3}\right),e\left(\frac{3}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4896 }(4333, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{5}{8}\right)\) |
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sage:chi.jacobi_sum(n)
