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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1899, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([70,66]))
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gp:[g,chi] = znchar(Mod(1120, 1899))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1899.1120");
| Modulus: | \(1899\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1899\) |
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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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\(\chi_{1899}(13,\cdot)\)
\(\chi_{1899}(25,\cdot)\)
\(\chi_{1899}(76,\cdot)\)
\(\chi_{1899}(79,\cdot)\)
\(\chi_{1899}(121,\cdot)\)
\(\chi_{1899}(151,\cdot)\)
\(\chi_{1899}(169,\cdot)\)
\(\chi_{1899}(184,\cdot)\)
\(\chi_{1899}(193,\cdot)\)
\(\chi_{1899}(394,\cdot)\)
\(\chi_{1899}(427,\cdot)\)
\(\chi_{1899}(535,\cdot)\)
\(\chi_{1899}(544,\cdot)\)
\(\chi_{1899}(547,\cdot)\)
\(\chi_{1899}(565,\cdot)\)
\(\chi_{1899}(625,\cdot)\)
\(\chi_{1899}(646,\cdot)\)
\(\chi_{1899}(697,\cdot)\)
\(\chi_{1899}(709,\cdot)\)
\(\chi_{1899}(715,\cdot)\)
\(\chi_{1899}(742,\cdot)\)
\(\chi_{1899}(754,\cdot)\)
\(\chi_{1899}(817,\cdot)\)
\(\chi_{1899}(826,\cdot)\)
\(\chi_{1899}(931,\cdot)\)
\(\chi_{1899}(940,\cdot)\)
\(\chi_{1899}(958,\cdot)\)
\(\chi_{1899}(1060,\cdot)\)
\(\chi_{1899}(1066,\cdot)\)
\(\chi_{1899}(1120,\cdot)\)
...
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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 };
\((1478,424)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{11}{35}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1899 }(1120, a) \) |
\(1\) | \(1\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{62}{105}\right)\) |
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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)
