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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(28861, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([145,105,84]))
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gp:[g,chi] = znchar(Mod(11912, 28861))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("28861.11912");
| Modulus: | \(28861\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(28861\) |
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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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\(\chi_{28861}(1025,\cdot)\)
\(\chi_{28861}(1690,\cdot)\)
\(\chi_{28861}(2203,\cdot)\)
\(\chi_{28861}(2488,\cdot)\)
\(\chi_{28861}(2868,\cdot)\)
\(\chi_{28861}(2887,\cdot)\)
\(\chi_{28861}(3666,\cdot)\)
\(\chi_{28861}(4065,\cdot)\)
\(\chi_{28861}(5148,\cdot)\)
\(\chi_{28861}(6326,\cdot)\)
\(\chi_{28861}(6611,\cdot)\)
\(\chi_{28861}(6991,\cdot)\)
\(\chi_{28861}(7010,\cdot)\)
\(\chi_{28861}(7789,\cdot)\)
\(\chi_{28861}(8188,\cdot)\)
\(\chi_{28861}(9271,\cdot)\)
\(\chi_{28861}(9936,\cdot)\)
\(\chi_{28861}(10449,\cdot)\)
\(\chi_{28861}(10734,\cdot)\)
\(\chi_{28861}(11114,\cdot)\)
\(\chi_{28861}(11133,\cdot)\)
\(\chi_{28861}(11912,\cdot)\)
\(\chi_{28861}(12311,\cdot)\)
\(\chi_{28861}(13394,\cdot)\)
\(\chi_{28861}(14059,\cdot)\)
\(\chi_{28861}(14857,\cdot)\)
\(\chi_{28861}(15237,\cdot)\)
\(\chi_{28861}(15256,\cdot)\)
\(\chi_{28861}(16035,\cdot)\)
\(\chi_{28861}(17517,\cdot)\)
...
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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 };
\((1179,27343,1863)\) → \((e\left(\frac{29}{42}\right),-1,e\left(\frac{2}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 28861 }(11912, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{210}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{73}{105}\right)\) |
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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)
