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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4255, base_ring=CyclotomicField(198))
M = H._module
chi = DirichletCharacter(H, M([99,18,187]))
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gp:[g,chi] = znchar(Mod(324, 4255))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4255.324");
| Modulus: | \(4255\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4255\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(198\) |
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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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\(\chi_{4255}(4,\cdot)\)
\(\chi_{4255}(104,\cdot)\)
\(\chi_{4255}(169,\cdot)\)
\(\chi_{4255}(284,\cdot)\)
\(\chi_{4255}(289,\cdot)\)
\(\chi_{4255}(324,\cdot)\)
\(\chi_{4255}(354,\cdot)\)
\(\chi_{4255}(374,\cdot)\)
\(\chi_{4255}(469,\cdot)\)
\(\chi_{4255}(509,\cdot)\)
\(\chi_{4255}(669,\cdot)\)
\(\chi_{4255}(694,\cdot)\)
\(\chi_{4255}(744,\cdot)\)
\(\chi_{4255}(844,\cdot)\)
\(\chi_{4255}(854,\cdot)\)
\(\chi_{4255}(909,\cdot)\)
\(\chi_{4255}(929,\cdot)\)
\(\chi_{4255}(1024,\cdot)\)
\(\chi_{4255}(1039,\cdot)\)
\(\chi_{4255}(1064,\cdot)\)
\(\chi_{4255}(1094,\cdot)\)
\(\chi_{4255}(1209,\cdot)\)
\(\chi_{4255}(1214,\cdot)\)
\(\chi_{4255}(1409,\cdot)\)
\(\chi_{4255}(1434,\cdot)\)
\(\chi_{4255}(1484,\cdot)\)
\(\chi_{4255}(1619,\cdot)\)
\(\chi_{4255}(1649,\cdot)\)
\(\chi_{4255}(1669,\cdot)\)
\(\chi_{4255}(1764,\cdot)\)
...
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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 };
\((852,741,3221)\) → \((-1,e\left(\frac{1}{11}\right),e\left(\frac{17}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 4255 }(324, a) \) |
\(1\) | \(1\) | \(e\left(\frac{62}{99}\right)\) | \(e\left(\frac{101}{198}\right)\) | \(e\left(\frac{25}{99}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{89}{198}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{2}{99}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{151}{198}\right)\) | \(e\left(\frac{16}{99}\right)\) |
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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)
