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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11687, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([105,165,119]))
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gp:[g,chi] = znchar(Mod(5602, 11687))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11687.5602");
| Modulus: | \(11687\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(11687\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(210\) |
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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: | odd |
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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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\(\chi_{11687}(207,\cdot)\)
\(\chi_{11687}(415,\cdot)\)
\(\chi_{11687}(792,\cdot)\)
\(\chi_{11687}(818,\cdot)\)
\(\chi_{11687}(1078,\cdot)\)
\(\chi_{11687}(1169,\cdot)\)
\(\chi_{11687}(1195,\cdot)\)
\(\chi_{11687}(1572,\cdot)\)
\(\chi_{11687}(1832,\cdot)\)
\(\chi_{11687}(2586,\cdot)\)
\(\chi_{11687}(2677,\cdot)\)
\(\chi_{11687}(2690,\cdot)\)
\(\chi_{11687}(3080,\cdot)\)
\(\chi_{11687}(3431,\cdot)\)
\(\chi_{11687}(3444,\cdot)\)
\(\chi_{11687}(3834,\cdot)\)
\(\chi_{11687}(3899,\cdot)\)
\(\chi_{11687}(4198,\cdot)\)
\(\chi_{11687}(4653,\cdot)\)
\(\chi_{11687}(4848,\cdot)\)
\(\chi_{11687}(5108,\cdot)\)
\(\chi_{11687}(5225,\cdot)\)
\(\chi_{11687}(5407,\cdot)\)
\(\chi_{11687}(5602,\cdot)\)
\(\chi_{11687}(5862,\cdot)\)
\(\chi_{11687}(6460,\cdot)\)
\(\chi_{11687}(6616,\cdot)\)
\(\chi_{11687}(6837,\cdot)\)
\(\chi_{11687}(7110,\cdot)\)
\(\chi_{11687}(7214,\cdot)\)
...
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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 };
\((6294,7658,7164)\) → \((-1,e\left(\frac{11}{14}\right),e\left(\frac{17}{30}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 11687 }(5602, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{167}{210}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{37}{210}\right)\) |
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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)
