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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(26299, base_ring=CyclotomicField(102))
M = H._module
chi = DirichletCharacter(H, M([51,34,69]))
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gp:[g,chi] = znchar(Mod(6425, 26299))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("26299.6425");
| Modulus: | \(26299\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(26299\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(102\) |
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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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\(\chi_{26299}(237,\cdot)\)
\(\chi_{26299}(594,\cdot)\)
\(\chi_{26299}(1784,\cdot)\)
\(\chi_{26299}(2141,\cdot)\)
\(\chi_{26299}(3331,\cdot)\)
\(\chi_{26299}(3688,\cdot)\)
\(\chi_{26299}(4878,\cdot)\)
\(\chi_{26299}(5235,\cdot)\)
\(\chi_{26299}(6425,\cdot)\)
\(\chi_{26299}(6782,\cdot)\)
\(\chi_{26299}(7972,\cdot)\)
\(\chi_{26299}(8329,\cdot)\)
\(\chi_{26299}(9519,\cdot)\)
\(\chi_{26299}(9876,\cdot)\)
\(\chi_{26299}(11066,\cdot)\)
\(\chi_{26299}(11423,\cdot)\)
\(\chi_{26299}(12613,\cdot)\)
\(\chi_{26299}(12970,\cdot)\)
\(\chi_{26299}(14517,\cdot)\)
\(\chi_{26299}(15707,\cdot)\)
\(\chi_{26299}(16064,\cdot)\)
\(\chi_{26299}(17254,\cdot)\)
\(\chi_{26299}(17611,\cdot)\)
\(\chi_{26299}(18801,\cdot)\)
\(\chi_{26299}(19158,\cdot)\)
\(\chi_{26299}(20348,\cdot)\)
\(\chi_{26299}(20705,\cdot)\)
\(\chi_{26299}(21895,\cdot)\)
\(\chi_{26299}(23442,\cdot)\)
\(\chi_{26299}(23799,\cdot)\)
...
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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 };
\((22543,10116,9829)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{23}{34}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 26299 }(6425, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{14}{51}\right)\) | \(e\left(\frac{91}{102}\right)\) | \(e\left(\frac{4}{17}\right)\) |
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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)
