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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(152269, base_ring=CyclotomicField(312))
M = H._module
chi = DirichletCharacter(H, M([200,117,36]))
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gp:[g,chi] = znchar(Mod(27147, 152269))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("152269.27147");
| 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: | \(312\) |
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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);
|
\(\chi_{152269}(536,\cdot)\)
\(\chi_{152269}(620,\cdot)\)
\(\chi_{152269}(835,\cdot)\)
\(\chi_{152269}(3136,\cdot)\)
\(\chi_{152269}(5976,\cdot)\)
\(\chi_{152269}(7692,\cdot)\)
\(\chi_{152269}(8570,\cdot)\)
\(\chi_{152269}(9233,\cdot)\)
\(\chi_{152269}(9512,\cdot)\)
\(\chi_{152269}(9597,\cdot)\)
\(\chi_{152269}(9811,\cdot)\)
\(\chi_{152269}(10344,\cdot)\)
\(\chi_{152269}(13575,\cdot)\)
\(\chi_{152269}(15122,\cdot)\)
\(\chi_{152269}(18749,\cdot)\)
\(\chi_{152269}(20952,\cdot)\)
\(\chi_{152269}(25151,\cdot)\)
\(\chi_{152269}(25730,\cdot)\)
\(\chi_{152269}(26484,\cdot)\)
\(\chi_{152269}(27147,\cdot)\)
\(\chi_{152269}(27491,\cdot)\)
\(\chi_{152269}(27725,\cdot)\)
\(\chi_{152269}(30007,\cdot)\)
\(\chi_{152269}(34563,\cdot)\)
\(\chi_{152269}(36507,\cdot)\)
\(\chi_{152269}(37215,\cdot)\)
\(\chi_{152269}(38853,\cdot)\)
\(\chi_{152269}(45405,\cdot)\)
\(\chi_{152269}(45620,\cdot)\)
\(\chi_{152269}(47823,\cdot)\)
...
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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 };
\((149567,143313,83318)\) → \((e\left(\frac{25}{39}\right),e\left(\frac{3}{8}\right),e\left(\frac{3}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 152269 }(27147, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{156}\right)\) | \(e\left(\frac{257}{312}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{7}{104}\right)\) | \(e\left(\frac{259}{312}\right)\) | \(e\left(\frac{103}{312}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{101}{156}\right)\) | \(e\left(\frac{23}{312}\right)\) | \(e\left(\frac{107}{312}\right)\) |
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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)
