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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14663, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([147,196,195]))
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gp:[g,chi] = znchar(Mod(4781, 14663))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("14663.4781");
| Modulus: | \(14663\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(14663\) |
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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: | 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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\(\chi_{14663}(51,\cdot)\)
\(\chi_{14663}(204,\cdot)\)
\(\chi_{14663}(475,\cdot)\)
\(\chi_{14663}(710,\cdot)\)
\(\chi_{14663}(733,\cdot)\)
\(\chi_{14663}(1157,\cdot)\)
\(\chi_{14663}(1415,\cdot)\)
\(\chi_{14663}(1900,\cdot)\)
\(\chi_{14663}(2053,\cdot)\)
\(\chi_{14663}(2158,\cdot)\)
\(\chi_{14663}(2521,\cdot)\)
\(\chi_{14663}(2582,\cdot)\)
\(\chi_{14663}(2779,\cdot)\)
\(\chi_{14663}(2840,\cdot)\)
\(\chi_{14663}(2932,\cdot)\)
\(\chi_{14663}(3264,\cdot)\)
\(\chi_{14663}(3522,\cdot)\)
\(\chi_{14663}(3614,\cdot)\)
\(\chi_{14663}(4296,\cdot)\)
\(\chi_{14663}(4461,\cdot)\)
\(\chi_{14663}(4628,\cdot)\)
\(\chi_{14663}(4781,\cdot)\)
\(\chi_{14663}(4881,\cdot)\)
\(\chi_{14663}(4886,\cdot)\)
\(\chi_{14663}(5463,\cdot)\)
\(\chi_{14663}(5660,\cdot)\)
\(\chi_{14663}(6145,\cdot)\)
\(\chi_{14663}(7189,\cdot)\)
\(\chi_{14663}(7509,\cdot)\)
\(\chi_{14663}(7871,\cdot)\)
...
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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 };
\((7999,3785,9549)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{14}{15}\right),e\left(\frac{13}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 14663 }(4781, a) \) |
\(1\) | \(1\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{97}{210}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{143}{210}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{179}{210}\right)\) | \(e\left(\frac{169}{210}\right)\) |
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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)
