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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4689, base_ring=CyclotomicField(312))
M = H._module
chi = DirichletCharacter(H, M([104,213]))
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gp:[g,chi] = znchar(Mod(1903, 4689))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4689.1903");
| Modulus: | \(4689\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4689\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(312\) |
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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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\(\chi_{4689}(34,\cdot)\)
\(\chi_{4689}(61,\cdot)\)
\(\chi_{4689}(175,\cdot)\)
\(\chi_{4689}(187,\cdot)\)
\(\chi_{4689}(205,\cdot)\)
\(\chi_{4689}(214,\cdot)\)
\(\chi_{4689}(229,\cdot)\)
\(\chi_{4689}(247,\cdot)\)
\(\chi_{4689}(268,\cdot)\)
\(\chi_{4689}(274,\cdot)\)
\(\chi_{4689}(292,\cdot)\)
\(\chi_{4689}(340,\cdot)\)
\(\chi_{4689}(346,\cdot)\)
\(\chi_{4689}(430,\cdot)\)
\(\chi_{4689}(475,\cdot)\)
\(\chi_{4689}(610,\cdot)\)
\(\chi_{4689}(652,\cdot)\)
\(\chi_{4689}(673,\cdot)\)
\(\chi_{4689}(799,\cdot)\)
\(\chi_{4689}(907,\cdot)\)
\(\chi_{4689}(967,\cdot)\)
\(\chi_{4689}(1003,\cdot)\)
\(\chi_{4689}(1066,\cdot)\)
\(\chi_{4689}(1174,\cdot)\)
\(\chi_{4689}(1177,\cdot)\)
\(\chi_{4689}(1255,\cdot)\)
\(\chi_{4689}(1282,\cdot)\)
\(\chi_{4689}(1285,\cdot)\)
\(\chi_{4689}(1411,\cdot)\)
\(\chi_{4689}(1474,\cdot)\)
...
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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 };
\((4169,1045)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{71}{104}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 4689 }(1903, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{67}{156}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{263}{312}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{121}{156}\right)\) | \(e\left(\frac{59}{156}\right)\) | \(e\left(\frac{85}{312}\right)\) | \(e\left(\frac{28}{39}\right)\) |
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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)
