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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1859, base_ring=CyclotomicField(130))
M = H._module
chi = DirichletCharacter(H, M([52,80]))
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gp:[g,chi] = znchar(Mod(313, 1859))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1859.313");
| Modulus: | \(1859\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1859\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(65\) |
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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);
|
\(\chi_{1859}(14,\cdot)\)
\(\chi_{1859}(27,\cdot)\)
\(\chi_{1859}(53,\cdot)\)
\(\chi_{1859}(92,\cdot)\)
\(\chi_{1859}(157,\cdot)\)
\(\chi_{1859}(196,\cdot)\)
\(\chi_{1859}(235,\cdot)\)
\(\chi_{1859}(300,\cdot)\)
\(\chi_{1859}(313,\cdot)\)
\(\chi_{1859}(378,\cdot)\)
\(\chi_{1859}(443,\cdot)\)
\(\chi_{1859}(456,\cdot)\)
\(\chi_{1859}(482,\cdot)\)
\(\chi_{1859}(521,\cdot)\)
\(\chi_{1859}(586,\cdot)\)
\(\chi_{1859}(599,\cdot)\)
\(\chi_{1859}(625,\cdot)\)
\(\chi_{1859}(664,\cdot)\)
\(\chi_{1859}(729,\cdot)\)
\(\chi_{1859}(742,\cdot)\)
\(\chi_{1859}(768,\cdot)\)
\(\chi_{1859}(807,\cdot)\)
\(\chi_{1859}(872,\cdot)\)
\(\chi_{1859}(885,\cdot)\)
\(\chi_{1859}(911,\cdot)\)
\(\chi_{1859}(950,\cdot)\)
\(\chi_{1859}(1028,\cdot)\)
\(\chi_{1859}(1054,\cdot)\)
\(\chi_{1859}(1093,\cdot)\)
\(\chi_{1859}(1158,\cdot)\)
...
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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 };
\((508,1354)\) → \((e\left(\frac{2}{5}\right),e\left(\frac{8}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 1859 }(313, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) |
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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)
