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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(80465, base_ring=CyclotomicField(396))
M = H._module
chi = DirichletCharacter(H, M([99,132,72,44]))
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gp:[g,chi] = znchar(Mod(11727, 80465))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("80465.11727");
| Modulus: | \(80465\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(80465\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(396\) |
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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);
|
\(\chi_{80465}(23,\cdot)\)
\(\chi_{80465}(177,\cdot)\)
\(\chi_{80465}(408,\cdot)\)
\(\chi_{80465}(1992,\cdot)\)
\(\chi_{80465}(2608,\cdot)\)
\(\chi_{80465}(3103,\cdot)\)
\(\chi_{80465}(3532,\cdot)\)
\(\chi_{80465}(4412,\cdot)\)
\(\chi_{80465}(4797,\cdot)\)
\(\chi_{80465}(4918,\cdot)\)
\(\chi_{80465}(6458,\cdot)\)
\(\chi_{80465}(6997,\cdot)\)
\(\chi_{80465}(7338,\cdot)\)
\(\chi_{80465}(7492,\cdot)\)
\(\chi_{80465}(7723,\cdot)\)
\(\chi_{80465}(9307,\cdot)\)
\(\chi_{80465}(10418,\cdot)\)
\(\chi_{80465}(10847,\cdot)\)
\(\chi_{80465}(11727,\cdot)\)
\(\chi_{80465}(12112,\cdot)\)
\(\chi_{80465}(12233,\cdot)\)
\(\chi_{80465}(13773,\cdot)\)
\(\chi_{80465}(14312,\cdot)\)
\(\chi_{80465}(14653,\cdot)\)
\(\chi_{80465}(14807,\cdot)\)
\(\chi_{80465}(15038,\cdot)\)
\(\chi_{80465}(16622,\cdot)\)
\(\chi_{80465}(17238,\cdot)\)
\(\chi_{80465}(17733,\cdot)\)
\(\chi_{80465}(18162,\cdot)\)
...
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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 };
\((32187,22991,79136,38116)\) → \((i,e\left(\frac{1}{3}\right),e\left(\frac{2}{11}\right),e\left(\frac{1}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
| \( \chi_{ 80465 }(11727, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{83}{396}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{83}{198}\right)\) | \(e\left(\frac{73}{99}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{265}{396}\right)\) | \(e\left(\frac{83}{99}\right)\) | \(e\left(\frac{239}{396}\right)\) |
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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)
