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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13351, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([75,70]))
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gp:[g,chi] = znchar(Mod(2725, 13351))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13351.2725");
| Modulus: | \(13351\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(13351\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(156\) |
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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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\(\chi_{13351}(307,\cdot)\)
\(\chi_{13351}(2556,\cdot)\)
\(\chi_{13351}(2725,\cdot)\)
\(\chi_{13351}(2878,\cdot)\)
\(\chi_{13351}(2891,\cdot)\)
\(\chi_{13351}(3050,\cdot)\)
\(\chi_{13351}(3307,\cdot)\)
\(\chi_{13351}(3463,\cdot)\)
\(\chi_{13351}(3671,\cdot)\)
\(\chi_{13351}(3827,\cdot)\)
\(\chi_{13351}(4373,\cdot)\)
\(\chi_{13351}(4542,\cdot)\)
\(\chi_{13351}(4815,\cdot)\)
\(\chi_{13351}(5426,\cdot)\)
\(\chi_{13351}(5741,\cdot)\)
\(\chi_{13351}(5923,\cdot)\)
\(\chi_{13351}(5962,\cdot)\)
\(\chi_{13351}(5988,\cdot)\)
\(\chi_{13351}(6921,\cdot)\)
\(\chi_{13351}(7298,\cdot)\)
\(\chi_{13351}(7311,\cdot)\)
\(\chi_{13351}(7376,\cdot)\)
\(\chi_{13351}(8172,\cdot)\)
\(\chi_{13351}(8263,\cdot)\)
\(\chi_{13351}(8575,\cdot)\)
\(\chi_{13351}(9001,\cdot)\)
\(\chi_{13351}(9092,\cdot)\)
\(\chi_{13351}(9329,\cdot)\)
\(\chi_{13351}(9407,\cdot)\)
\(\chi_{13351}(9706,\cdot)\)
...
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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 };
\((7269,12169)\) → \((e\left(\frac{25}{52}\right),e\left(\frac{35}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 13351 }(2725, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{156}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{23}{156}\right)\) | \(e\left(\frac{53}{156}\right)\) | \(e\left(\frac{35}{156}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{5}{156}\right)\) |
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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)
