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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1763, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([168,25]))
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gp:[g,chi] = znchar(Mod(1103, 1763))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1763.1103");
| Modulus: | \(1763\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1763\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(210\) |
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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);
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\(\chi_{1763}(18,\cdot)\)
\(\chi_{1763}(98,\cdot)\)
\(\chi_{1763}(119,\cdot)\)
\(\chi_{1763}(141,\cdot)\)
\(\chi_{1763}(201,\cdot)\)
\(\chi_{1763}(406,\cdot)\)
\(\chi_{1763}(420,\cdot)\)
\(\chi_{1763}(502,\cdot)\)
\(\chi_{1763}(549,\cdot)\)
\(\chi_{1763}(592,\cdot)\)
\(\chi_{1763}(631,\cdot)\)
\(\chi_{1763}(674,\cdot)\)
\(\chi_{1763}(693,\cdot)\)
\(\chi_{1763}(707,\cdot)\)
\(\chi_{1763}(734,\cdot)\)
\(\chi_{1763}(836,\cdot)\)
\(\chi_{1763}(879,\cdot)\)
\(\chi_{1763}(980,\cdot)\)
\(\chi_{1763}(994,\cdot)\)
\(\chi_{1763}(1035,\cdot)\)
\(\chi_{1763}(1062,\cdot)\)
\(\chi_{1763}(1103,\cdot)\)
\(\chi_{1763}(1123,\cdot)\)
\(\chi_{1763}(1144,\cdot)\)
\(\chi_{1763}(1164,\cdot)\)
\(\chi_{1763}(1166,\cdot)\)
\(\chi_{1763}(1207,\cdot)\)
\(\chi_{1763}(1267,\cdot)\)
\(\chi_{1763}(1281,\cdot)\)
\(\chi_{1763}(1308,\cdot)\)
...
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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 };
\((990,1723)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{5}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1763 }(1103, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{121}{210}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{34}{35}\right)\) |
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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)
