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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(152269, base_ring=CyclotomicField(624))
M = H._module
chi = DirichletCharacter(H, M([172,507,372]))
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gp:[g,chi] = znchar(Mod(2936, 152269))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("152269.2936");
| Modulus: | \(152269\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(152269\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(624\) |
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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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\(\chi_{152269}(1744,\cdot)\)
\(\chi_{152269}(1814,\cdot)\)
\(\chi_{152269}(2034,\cdot)\)
\(\chi_{152269}(2645,\cdot)\)
\(\chi_{152269}(2936,\cdot)\)
\(\chi_{152269}(3802,\cdot)\)
\(\chi_{152269}(5531,\cdot)\)
\(\chi_{152269}(5553,\cdot)\)
\(\chi_{152269}(6312,\cdot)\)
\(\chi_{152269}(7135,\cdot)\)
\(\chi_{152269}(7213,\cdot)\)
\(\chi_{152269}(7772,\cdot)\)
\(\chi_{152269}(8123,\cdot)\)
\(\chi_{152269}(10142,\cdot)\)
\(\chi_{152269}(12434,\cdot)\)
\(\chi_{152269}(13427,\cdot)\)
\(\chi_{152269}(13648,\cdot)\)
\(\chi_{152269}(15269,\cdot)\)
\(\chi_{152269}(16170,\cdot)\)
\(\chi_{152269}(17080,\cdot)\)
\(\chi_{152269}(19658,\cdot)\)
\(\chi_{152269}(19948,\cdot)\)
\(\chi_{152269}(20559,\cdot)\)
\(\chi_{152269}(20720,\cdot)\)
\(\chi_{152269}(21340,\cdot)\)
\(\chi_{152269}(21703,\cdot)\)
\(\chi_{152269}(21716,\cdot)\)
\(\chi_{152269}(22384,\cdot)\)
\(\chi_{152269}(22605,\cdot)\)
\(\chi_{152269}(23467,\cdot)\)
...
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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 };
\((149567,143313,83318)\) → \((e\left(\frac{43}{156}\right),e\left(\frac{13}{16}\right),e\left(\frac{31}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 152269 }(2936, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{77}{312}\right)\) | \(e\left(\frac{79}{624}\right)\) | \(e\left(\frac{77}{156}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{233}{624}\right)\) | \(e\left(\frac{485}{624}\right)\) | \(e\left(\frac{77}{104}\right)\) | \(e\left(\frac{79}{312}\right)\) | \(e\left(\frac{505}{624}\right)\) | \(e\left(\frac{409}{624}\right)\) |
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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)
