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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5141, base_ring=CyclotomicField(208))
M = H._module
chi = DirichletCharacter(H, M([100,169]))
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gp:[g,chi] = znchar(Mod(503, 5141))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5141.503");
| Modulus: | \(5141\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5141\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(208\) |
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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);
|
\(\chi_{5141}(12,\cdot)\)
\(\chi_{5141}(27,\cdot)\)
\(\chi_{5141}(79,\cdot)\)
\(\chi_{5141}(124,\cdot)\)
\(\chi_{5141}(167,\cdot)\)
\(\chi_{5141}(279,\cdot)\)
\(\chi_{5141}(376,\cdot)\)
\(\chi_{5141}(406,\cdot)\)
\(\chi_{5141}(458,\cdot)\)
\(\chi_{5141}(503,\cdot)\)
\(\chi_{5141}(671,\cdot)\)
\(\chi_{5141}(764,\cdot)\)
\(\chi_{5141}(768,\cdot)\)
\(\chi_{5141}(803,\cdot)\)
\(\chi_{5141}(846,\cdot)\)
\(\chi_{5141}(1040,\cdot)\)
\(\chi_{5141}(1094,\cdot)\)
\(\chi_{5141}(1346,\cdot)\)
\(\chi_{5141}(1366,\cdot)\)
\(\chi_{5141}(1428,\cdot)\)
\(\chi_{5141}(1443,\cdot)\)
\(\chi_{5141}(1463,\cdot)\)
\(\chi_{5141}(1482,\cdot)\)
\(\chi_{5141}(1564,\cdot)\)
\(\chi_{5141}(1631,\cdot)\)
\(\chi_{5141}(1661,\cdot)\)
\(\chi_{5141}(1676,\cdot)\)
\(\chi_{5141}(1728,\cdot)\)
\(\chi_{5141}(1816,\cdot)\)
\(\chi_{5141}(1913,\cdot)\)
...
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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 };
\((4560,2333)\) → \((e\left(\frac{25}{52}\right),e\left(\frac{13}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 5141 }(503, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{104}\right)\) | \(e\left(\frac{5}{104}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{85}{208}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{191}{208}\right)\) | \(e\left(\frac{33}{104}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{107}{208}\right)\) | \(e\left(\frac{79}{104}\right)\) |
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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)
