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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2380, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,36,40,27]))
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pari:[g,chi] = znchar(Mod(2343,2380))
| Modulus: | \(2380\) | |
| Conductor: | \(2380\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(48\) |
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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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\(\chi_{2380}(3,\cdot)\)
\(\chi_{2380}(227,\cdot)\)
\(\chi_{2380}(243,\cdot)\)
\(\chi_{2380}(367,\cdot)\)
\(\chi_{2380}(983,\cdot)\)
\(\chi_{2380}(1027,\cdot)\)
\(\chi_{2380}(1083,\cdot)\)
\(\chi_{2380}(1167,\cdot)\)
\(\chi_{2380}(1263,\cdot)\)
\(\chi_{2380}(1363,\cdot)\)
\(\chi_{2380}(1587,\cdot)\)
\(\chi_{2380}(1727,\cdot)\)
\(\chi_{2380}(2047,\cdot)\)
\(\chi_{2380}(2103,\cdot)\)
\(\chi_{2380}(2187,\cdot)\)
\(\chi_{2380}(2343,\cdot)\)
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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 ]
\((1191,477,1361,1261)\) → \((-1,-i,e\left(\frac{5}{6}\right),e\left(\frac{9}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(33\) |
| \( \chi_{ 2380 }(2343, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{11}{12}\right)\) |
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sage:chi.jacobi_sum(n)
