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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2380, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,12,4,15]))
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pari:[g,chi] = znchar(Mod(59,2380))
| Modulus: | \(2380\) | |
| Conductor: | \(2380\) |
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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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\(\chi_{2380}(19,\cdot)\)
\(\chi_{2380}(59,\cdot)\)
\(\chi_{2380}(859,\cdot)\)
\(\chi_{2380}(899,\cdot)\)
\(\chi_{2380}(1039,\cdot)\)
\(\chi_{2380}(1419,\cdot)\)
\(\chi_{2380}(1879,\cdot)\)
\(\chi_{2380}(2259,\cdot)\)
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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 ]
\((1191,477,1361,1261)\) → \((-1,-1,e\left(\frac{1}{6}\right),e\left(\frac{5}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(33\) |
| \( \chi_{ 2380 }(59, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(-1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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sage:chi.jacobi_sum(n)
