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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(23177, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([115,168,140]))
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gp:[g,chi] = znchar(Mod(9595, 23177))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("23177.9595");
| Modulus: | \(23177\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(23177\) |
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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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\(\chi_{23177}(565,\cdot)\)
\(\chi_{23177}(724,\cdot)\)
\(\chi_{23177}(1468,\cdot)\)
\(\chi_{23177}(1769,\cdot)\)
\(\chi_{23177}(1928,\cdot)\)
\(\chi_{23177}(2831,\cdot)\)
\(\chi_{23177}(2973,\cdot)\)
\(\chi_{23177}(3734,\cdot)\)
\(\chi_{23177}(3876,\cdot)\)
\(\chi_{23177}(4035,\cdot)\)
\(\chi_{23177}(4779,\cdot)\)
\(\chi_{23177}(5080,\cdot)\)
\(\chi_{23177}(5239,\cdot)\)
\(\chi_{23177}(6142,\cdot)\)
\(\chi_{23177}(6284,\cdot)\)
\(\chi_{23177}(7045,\cdot)\)
\(\chi_{23177}(7187,\cdot)\)
\(\chi_{23177}(7346,\cdot)\)
\(\chi_{23177}(8090,\cdot)\)
\(\chi_{23177}(8391,\cdot)\)
\(\chi_{23177}(8550,\cdot)\)
\(\chi_{23177}(9453,\cdot)\)
\(\chi_{23177}(9595,\cdot)\)
\(\chi_{23177}(10356,\cdot)\)
\(\chi_{23177}(10498,\cdot)\)
\(\chi_{23177}(10657,\cdot)\)
\(\chi_{23177}(11401,\cdot)\)
\(\chi_{23177}(11702,\cdot)\)
\(\chi_{23177}(11861,\cdot)\)
\(\chi_{23177}(12764,\cdot)\)
...
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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 };
\((21759,4215,16171)\) → \((e\left(\frac{23}{42}\right),e\left(\frac{4}{5}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 23177 }(9595, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{157}{210}\right)\) | \(e\left(\frac{137}{210}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{43}{210}\right)\) |
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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)
