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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(18463, base_ring=CyclotomicField(1494))
M = H._module
chi = DirichletCharacter(H, M([1162,963]))
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gp:[g,chi] = znchar(Mod(12, 18463))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("18463.12");
| Modulus: | \(18463\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(18463\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1494\) |
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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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\(\chi_{18463}(12,\cdot)\)
\(\chi_{18463}(83,\cdot)\)
\(\chi_{18463}(108,\cdot)\)
\(\chi_{18463}(192,\cdot)\)
\(\chi_{18463}(201,\cdot)\)
\(\chi_{18463}(231,\cdot)\)
\(\chi_{18463}(305,\cdot)\)
\(\chi_{18463}(312,\cdot)\)
\(\chi_{18463}(330,\cdot)\)
\(\chi_{18463}(403,\cdot)\)
\(\chi_{18463}(460,\cdot)\)
\(\chi_{18463}(490,\cdot)\)
\(\chi_{18463}(493,\cdot)\)
\(\chi_{18463}(551,\cdot)\)
\(\chi_{18463}(571,\cdot)\)
\(\chi_{18463}(700,\cdot)\)
\(\chi_{18463}(719,\cdot)\)
\(\chi_{18463}(747,\cdot)\)
\(\chi_{18463}(749,\cdot)\)
\(\chi_{18463}(789,\cdot)\)
\(\chi_{18463}(811,\cdot)\)
\(\chi_{18463}(821,\cdot)\)
\(\chi_{18463}(921,\cdot)\)
\(\chi_{18463}(934,\cdot)\)
\(\chi_{18463}(959,\cdot)\)
\(\chi_{18463}(974,\cdot)\)
\(\chi_{18463}(1006,\cdot)\)
\(\chi_{18463}(1011,\cdot)\)
\(\chi_{18463}(1070,\cdot)\)
\(\chi_{18463}(1106,\cdot)\)
...
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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 };
\((17466,5995)\) → \((e\left(\frac{7}{9}\right),e\left(\frac{107}{166}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 18463 }(12, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{253}{1494}\right)\) | \(e\left(\frac{521}{1494}\right)\) | \(e\left(\frac{253}{747}\right)\) | \(e\left(\frac{718}{747}\right)\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{797}{1494}\right)\) | \(e\left(\frac{253}{498}\right)\) | \(e\left(\frac{521}{747}\right)\) | \(e\left(\frac{65}{498}\right)\) | \(e\left(\frac{145}{498}\right)\) |
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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)
