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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6253, base_ring=CyclotomicField(234))
M = H._module
chi = DirichletCharacter(H, M([171,182]))
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gp:[g,chi] = znchar(Mod(2417, 6253))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6253.2417");
| Modulus: | \(6253\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(6253\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(234\) |
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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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\(\chi_{6253}(12,\cdot)\)
\(\chi_{6253}(90,\cdot)\)
\(\chi_{6253}(155,\cdot)\)
\(\chi_{6253}(181,\cdot)\)
\(\chi_{6253}(194,\cdot)\)
\(\chi_{6253}(441,\cdot)\)
\(\chi_{6253}(493,\cdot)\)
\(\chi_{6253}(571,\cdot)\)
\(\chi_{6253}(636,\cdot)\)
\(\chi_{6253}(662,\cdot)\)
\(\chi_{6253}(922,\cdot)\)
\(\chi_{6253}(974,\cdot)\)
\(\chi_{6253}(1052,\cdot)\)
\(\chi_{6253}(1117,\cdot)\)
\(\chi_{6253}(1143,\cdot)\)
\(\chi_{6253}(1156,\cdot)\)
\(\chi_{6253}(1403,\cdot)\)
\(\chi_{6253}(1455,\cdot)\)
\(\chi_{6253}(1533,\cdot)\)
\(\chi_{6253}(1598,\cdot)\)
\(\chi_{6253}(1624,\cdot)\)
\(\chi_{6253}(1637,\cdot)\)
\(\chi_{6253}(1884,\cdot)\)
\(\chi_{6253}(1936,\cdot)\)
\(\chi_{6253}(2014,\cdot)\)
\(\chi_{6253}(2079,\cdot)\)
\(\chi_{6253}(2105,\cdot)\)
\(\chi_{6253}(2118,\cdot)\)
\(\chi_{6253}(2417,\cdot)\)
\(\chi_{6253}(2495,\cdot)\)
...
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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 };
\((1185,5071)\) → \((e\left(\frac{19}{26}\right),e\left(\frac{7}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 6253 }(2417, a) \) |
\(1\) | \(1\) | \(e\left(\frac{119}{234}\right)\) | \(e\left(\frac{98}{117}\right)\) | \(e\left(\frac{2}{117}\right)\) | \(e\left(\frac{109}{234}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{19}{234}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{79}{117}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{47}{78}\right)\) |
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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)
