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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12384, base_ring=CyclotomicField(168))
M = H._module
chi = DirichletCharacter(H, M([84,21,140,72]))
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gp:[g,chi] = znchar(Mod(7259, 12384))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("12384.7259");
| Modulus: | \(12384\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(12384\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(168\) |
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sage:chi.multiplicative_order()
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gp:charorder(g,chi)
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magma:Order(chi);
|
| Real: | no |
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sage:chi.multiplicative_order() <= 2
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gp:charorder(g,chi) <= 2
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magma:Order(chi) le 2;
|
| Primitive: | yes |
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sage:chi.is_primitive()
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gp:#znconreyconductor(g,chi)==1
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magma:IsPrimitive(chi);
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
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\(\chi_{12384}(11,\cdot)\)
\(\chi_{12384}(59,\cdot)\)
\(\chi_{12384}(299,\cdot)\)
\(\chi_{12384}(563,\cdot)\)
\(\chi_{12384}(1067,\cdot)\)
\(\chi_{12384}(1091,\cdot)\)
\(\chi_{12384}(1139,\cdot)\)
\(\chi_{12384}(1595,\cdot)\)
\(\chi_{12384}(2075,\cdot)\)
\(\chi_{12384}(2099,\cdot)\)
\(\chi_{12384}(2171,\cdot)\)
\(\chi_{12384}(2363,\cdot)\)
\(\chi_{12384}(3107,\cdot)\)
\(\chi_{12384}(3155,\cdot)\)
\(\chi_{12384}(3395,\cdot)\)
\(\chi_{12384}(3659,\cdot)\)
\(\chi_{12384}(4163,\cdot)\)
\(\chi_{12384}(4187,\cdot)\)
\(\chi_{12384}(4235,\cdot)\)
\(\chi_{12384}(4691,\cdot)\)
\(\chi_{12384}(5171,\cdot)\)
\(\chi_{12384}(5195,\cdot)\)
\(\chi_{12384}(5267,\cdot)\)
\(\chi_{12384}(5459,\cdot)\)
\(\chi_{12384}(6203,\cdot)\)
\(\chi_{12384}(6251,\cdot)\)
\(\chi_{12384}(6491,\cdot)\)
\(\chi_{12384}(6755,\cdot)\)
\(\chi_{12384}(7259,\cdot)\)
\(\chi_{12384}(7283,\cdot)\)
...
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sage:chi.galois_orbit()
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gp:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
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magma:order := Order(chi);
{ chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
\((3871,4645,11009,6625)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{5}{6}\right),e\left(\frac{3}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 12384 }(7259, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{137}{168}\right)\) | \(e\left(\frac{43}{168}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{31}{42}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
