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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2432, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,3,16]))
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pari:[g,chi] = znchar(Mod(2051,2432))
| Modulus: | \(2432\) | |
| Conductor: | \(2432\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(32\) |
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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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\(\chi_{2432}(75,\cdot)\)
\(\chi_{2432}(227,\cdot)\)
\(\chi_{2432}(379,\cdot)\)
\(\chi_{2432}(531,\cdot)\)
\(\chi_{2432}(683,\cdot)\)
\(\chi_{2432}(835,\cdot)\)
\(\chi_{2432}(987,\cdot)\)
\(\chi_{2432}(1139,\cdot)\)
\(\chi_{2432}(1291,\cdot)\)
\(\chi_{2432}(1443,\cdot)\)
\(\chi_{2432}(1595,\cdot)\)
\(\chi_{2432}(1747,\cdot)\)
\(\chi_{2432}(1899,\cdot)\)
\(\chi_{2432}(2051,\cdot)\)
\(\chi_{2432}(2203,\cdot)\)
\(\chi_{2432}(2355,\cdot)\)
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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 ]
\((1407,2053,1921)\) → \((-1,e\left(\frac{3}{32}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 2432 }(2051, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) |
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sage:chi.jacobi_sum(n)
