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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(16709, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([95,84,119]))
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gp:[g,chi] = znchar(Mod(4889, 16709))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("16709.4889");
| Modulus: | \(16709\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(16709\) |
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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: | 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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\(\chi_{16709}(115,\cdot)\)
\(\chi_{16709}(544,\cdot)\)
\(\chi_{16709}(663,\cdot)\)
\(\chi_{16709}(724,\cdot)\)
\(\chi_{16709}(768,\cdot)\)
\(\chi_{16709}(1181,\cdot)\)
\(\chi_{16709}(2005,\cdot)\)
\(\chi_{16709}(2369,\cdot)\)
\(\chi_{16709}(2502,\cdot)\)
\(\chi_{16709}(2931,\cdot)\)
\(\chi_{16709}(3050,\cdot)\)
\(\chi_{16709}(3111,\cdot)\)
\(\chi_{16709}(3568,\cdot)\)
\(\chi_{16709}(4756,\cdot)\)
\(\chi_{16709}(4889,\cdot)\)
\(\chi_{16709}(5318,\cdot)\)
\(\chi_{16709}(5437,\cdot)\)
\(\chi_{16709}(5498,\cdot)\)
\(\chi_{16709}(5542,\cdot)\)
\(\chi_{16709}(5955,\cdot)\)
\(\chi_{16709}(6779,\cdot)\)
\(\chi_{16709}(7143,\cdot)\)
\(\chi_{16709}(7276,\cdot)\)
\(\chi_{16709}(7705,\cdot)\)
\(\chi_{16709}(7824,\cdot)\)
\(\chi_{16709}(7885,\cdot)\)
\(\chi_{16709}(7929,\cdot)\)
\(\chi_{16709}(8342,\cdot)\)
\(\chi_{16709}(9166,\cdot)\)
\(\chi_{16709}(9530,\cdot)\)
...
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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 };
\((16369,1520,14015)\) → \((e\left(\frac{19}{42}\right),e\left(\frac{2}{5}\right),e\left(\frac{17}{30}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 16709 }(4889, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{11}{210}\right)\) | \(e\left(\frac{103}{105}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{59}{105}\right)\) |
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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)
