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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(29792, base_ring=CyclotomicField(504))
M = H._module
chi = DirichletCharacter(H, M([0,315,36,476]))
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gp:[g,chi] = znchar(Mod(4437, 29792))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("29792.4437");
| Modulus: | \(29792\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(29792\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(504\) |
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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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\(\chi_{29792}(13,\cdot)\)
\(\chi_{29792}(181,\cdot)\)
\(\chi_{29792}(573,\cdot)\)
\(\chi_{29792}(629,\cdot)\)
\(\chi_{29792}(965,\cdot)\)
\(\chi_{29792}(1021,\cdot)\)
\(\chi_{29792}(1245,\cdot)\)
\(\chi_{29792}(1637,\cdot)\)
\(\chi_{29792}(1693,\cdot)\)
\(\chi_{29792}(2029,\cdot)\)
\(\chi_{29792}(2085,\cdot)\)
\(\chi_{29792}(2141,\cdot)\)
\(\chi_{29792}(2309,\cdot)\)
\(\chi_{29792}(2701,\cdot)\)
\(\chi_{29792}(2757,\cdot)\)
\(\chi_{29792}(3093,\cdot)\)
\(\chi_{29792}(3149,\cdot)\)
\(\chi_{29792}(3205,\cdot)\)
\(\chi_{29792}(3373,\cdot)\)
\(\chi_{29792}(3765,\cdot)\)
\(\chi_{29792}(4157,\cdot)\)
\(\chi_{29792}(4269,\cdot)\)
\(\chi_{29792}(4437,\cdot)\)
\(\chi_{29792}(4829,\cdot)\)
\(\chi_{29792}(4885,\cdot)\)
\(\chi_{29792}(5221,\cdot)\)
\(\chi_{29792}(5277,\cdot)\)
\(\chi_{29792}(5333,\cdot)\)
\(\chi_{29792}(5501,\cdot)\)
\(\chi_{29792}(5893,\cdot)\)
...
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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 };
\((9311,11173,3041,3137)\) → \((1,e\left(\frac{5}{8}\right),e\left(\frac{1}{14}\right),e\left(\frac{17}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 29792 }(4437, a) \) |
\(1\) | \(1\) | \(e\left(\frac{113}{504}\right)\) | \(e\left(\frac{407}{504}\right)\) | \(e\left(\frac{113}{252}\right)\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{229}{504}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{89}{252}\right)\) | \(e\left(\frac{155}{252}\right)\) | \(e\left(\frac{113}{168}\right)\) |
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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)
