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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9991, base_ring=CyclotomicField(816))
M = H._module
chi = DirichletCharacter(H, M([459,304]))
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gp:[g,chi] = znchar(Mod(186, 9991))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9991.186");
| Modulus: | \(9991\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(9991\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(816\) |
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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);
|
\(\chi_{9991}(18,\cdot)\)
\(\chi_{9991}(105,\cdot)\)
\(\chi_{9991}(186,\cdot)\)
\(\chi_{9991}(221,\cdot)\)
\(\chi_{9991}(264,\cdot)\)
\(\chi_{9991}(303,\cdot)\)
\(\chi_{9991}(361,\cdot)\)
\(\chi_{9991}(400,\cdot)\)
\(\chi_{9991}(406,\cdot)\)
\(\chi_{9991}(467,\cdot)\)
\(\chi_{9991}(503,\cdot)\)
\(\chi_{9991}(564,\cdot)\)
\(\chi_{9991}(570,\cdot)\)
\(\chi_{9991}(574,\cdot)\)
\(\chi_{9991}(667,\cdot)\)
\(\chi_{9991}(749,\cdot)\)
\(\chi_{9991}(784,\cdot)\)
\(\chi_{9991}(803,\cdot)\)
\(\chi_{9991}(865,\cdot)\)
\(\chi_{9991}(943,\cdot)\)
\(\chi_{9991}(952,\cdot)\)
\(\chi_{9991}(982,\cdot)\)
\(\chi_{9991}(1049,\cdot)\)
\(\chi_{9991}(1055,\cdot)\)
\(\chi_{9991}(1059,\cdot)\)
\(\chi_{9991}(1079,\cdot)\)
\(\chi_{9991}(1085,\cdot)\)
\(\chi_{9991}(1137,\cdot)\)
\(\chi_{9991}(1152,\cdot)\)
\(\chi_{9991}(1182,\cdot)\)
...
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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 };
\((3400,6597)\) → \((e\left(\frac{9}{16}\right),e\left(\frac{19}{51}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 9991 }(186, a) \) |
\(1\) | \(1\) | \(e\left(\frac{211}{408}\right)\) | \(e\left(\frac{123}{136}\right)\) | \(e\left(\frac{7}{204}\right)\) | \(e\left(\frac{763}{816}\right)\) | \(e\left(\frac{43}{102}\right)\) | \(e\left(\frac{757}{816}\right)\) | \(e\left(\frac{75}{136}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{123}{272}\right)\) | \(e\left(\frac{41}{408}\right)\) |
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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)
