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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3437, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([140,198]))
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gp:[g,chi] = znchar(Mod(704, 3437))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3437.704");
| Modulus: | \(3437\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3437\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(105\) |
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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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\(\chi_{3437}(114,\cdot)\)
\(\chi_{3437}(144,\cdot)\)
\(\chi_{3437}(221,\cdot)\)
\(\chi_{3437}(226,\cdot)\)
\(\chi_{3437}(305,\cdot)\)
\(\chi_{3437}(681,\cdot)\)
\(\chi_{3437}(688,\cdot)\)
\(\chi_{3437}(704,\cdot)\)
\(\chi_{3437}(723,\cdot)\)
\(\chi_{3437}(746,\cdot)\)
\(\chi_{3437}(891,\cdot)\)
\(\chi_{3437}(919,\cdot)\)
\(\chi_{3437}(1038,\cdot)\)
\(\chi_{3437}(1096,\cdot)\)
\(\chi_{3437}(1208,\cdot)\)
\(\chi_{3437}(1311,\cdot)\)
\(\chi_{3437}(1493,\cdot)\)
\(\chi_{3437}(1514,\cdot)\)
\(\chi_{3437}(1654,\cdot)\)
\(\chi_{3437}(1663,\cdot)\)
\(\chi_{3437}(1670,\cdot)\)
\(\chi_{3437}(1703,\cdot)\)
\(\chi_{3437}(1705,\cdot)\)
\(\chi_{3437}(1836,\cdot)\)
\(\chi_{3437}(1873,\cdot)\)
\(\chi_{3437}(1901,\cdot)\)
\(\chi_{3437}(1976,\cdot)\)
\(\chi_{3437}(2020,\cdot)\)
\(\chi_{3437}(2172,\cdot)\)
\(\chi_{3437}(2221,\cdot)\)
...
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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 };
\((983,2948)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{33}{35}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 3437 }(704, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{64}{105}\right)\) | \(e\left(\frac{14}{15}\right)\) |
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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)
