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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([116,273,42]))
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gp:[g,chi] = znchar(Mod(16611, 152269))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("152269.16611");
| 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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{152269}(270,\cdot)\)
\(\chi_{152269}(4775,\cdot)\)
\(\chi_{152269}(10514,\cdot)\)
\(\chi_{152269}(10566,\cdot)\)
\(\chi_{152269}(11307,\cdot)\)
\(\chi_{152269}(13693,\cdot)\)
\(\chi_{152269}(14278,\cdot)\)
\(\chi_{152269}(15298,\cdot)\)
\(\chi_{152269}(15604,\cdot)\)
\(\chi_{152269}(16611,\cdot)\)
\(\chi_{152269}(21473,\cdot)\)
\(\chi_{152269}(22975,\cdot)\)
\(\chi_{152269}(28994,\cdot)\)
\(\chi_{152269}(29221,\cdot)\)
\(\chi_{152269}(30313,\cdot)\)
\(\chi_{152269}(30404,\cdot)\)
\(\chi_{152269}(30983,\cdot)\)
\(\chi_{152269}(31951,\cdot)\)
\(\chi_{152269}(34525,\cdot)\)
\(\chi_{152269}(34954,\cdot)\)
\(\chi_{152269}(35929,\cdot)\)
\(\chi_{152269}(38139,\cdot)\)
\(\chi_{152269}(38360,\cdot)\)
\(\chi_{152269}(39238,\cdot)\)
\(\chi_{152269}(39387,\cdot)\)
\(\chi_{152269}(40434,\cdot)\)
\(\chi_{152269}(42995,\cdot)\)
\(\chi_{152269}(44022,\cdot)\)
\(\chi_{152269}(46537,\cdot)\)
\(\chi_{152269}(48227,\cdot)\)
...
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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 };
\((149567,143313,83318)\) → \((e\left(\frac{29}{78}\right),e\left(\frac{7}{8}\right),e\left(\frac{7}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 152269 }(16611, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{83}{312}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{5}{104}\right)\) | \(e\left(\frac{7}{312}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{83}{156}\right)\) | \(e\left(\frac{251}{312}\right)\) | \(e\left(\frac{71}{312}\right)\) |
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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)
