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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(22477, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([26,59,52]))
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gp:[g,chi] = znchar(Mod(2971, 22477))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("22477.2971");
| Modulus: | \(22477\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(22477\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(156\) |
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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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\(\chi_{22477}(45,\cdot)\)
\(\chi_{22477}(600,\cdot)\)
\(\chi_{22477}(999,\cdot)\)
\(\chi_{22477}(1242,\cdot)\)
\(\chi_{22477}(1774,\cdot)\)
\(\chi_{22477}(2329,\cdot)\)
\(\chi_{22477}(2728,\cdot)\)
\(\chi_{22477}(2971,\cdot)\)
\(\chi_{22477}(3503,\cdot)\)
\(\chi_{22477}(4058,\cdot)\)
\(\chi_{22477}(4457,\cdot)\)
\(\chi_{22477}(4700,\cdot)\)
\(\chi_{22477}(5232,\cdot)\)
\(\chi_{22477}(5787,\cdot)\)
\(\chi_{22477}(6186,\cdot)\)
\(\chi_{22477}(6429,\cdot)\)
\(\chi_{22477}(6961,\cdot)\)
\(\chi_{22477}(7915,\cdot)\)
\(\chi_{22477}(8158,\cdot)\)
\(\chi_{22477}(8690,\cdot)\)
\(\chi_{22477}(9245,\cdot)\)
\(\chi_{22477}(9644,\cdot)\)
\(\chi_{22477}(9887,\cdot)\)
\(\chi_{22477}(10419,\cdot)\)
\(\chi_{22477}(10974,\cdot)\)
\(\chi_{22477}(11373,\cdot)\)
\(\chi_{22477}(11616,\cdot)\)
\(\chi_{22477}(12148,\cdot)\)
\(\chi_{22477}(12703,\cdot)\)
\(\chi_{22477}(13345,\cdot)\)
...
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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 };
\((19267,8114,4733)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{59}{156}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 22477 }(2971, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{156}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{89}{156}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{97}{156}\right)\) | \(e\left(\frac{19}{39}\right)\) |
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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)
