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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5881, base_ring=CyclotomicField(588))
M = H._module
chi = DirichletCharacter(H, M([79]))
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gp:[g,chi] = znchar(Mod(1081, 5881))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5881.1081");
| Modulus: | \(5881\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5881\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(588\) |
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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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\(\chi_{5881}(12,\cdot)\)
\(\chi_{5881}(23,\cdot)\)
\(\chi_{5881}(28,\cdot)\)
\(\chi_{5881}(41,\cdot)\)
\(\chi_{5881}(97,\cdot)\)
\(\chi_{5881}(126,\cdot)\)
\(\chi_{5881}(150,\cdot)\)
\(\chi_{5881}(210,\cdot)\)
\(\chi_{5881}(233,\cdot)\)
\(\chi_{5881}(243,\cdot)\)
\(\chi_{5881}(250,\cdot)\)
\(\chi_{5881}(306,\cdot)\)
\(\chi_{5881}(341,\cdot)\)
\(\chi_{5881}(361,\cdot)\)
\(\chi_{5881}(389,\cdot)\)
\(\chi_{5881}(446,\cdot)\)
\(\chi_{5881}(490,\cdot)\)
\(\chi_{5881}(494,\cdot)\)
\(\chi_{5881}(501,\cdot)\)
\(\chi_{5881}(510,\cdot)\)
\(\chi_{5881}(562,\cdot)\)
\(\chi_{5881}(564,\cdot)\)
\(\chi_{5881}(567,\cdot)\)
\(\chi_{5881}(640,\cdot)\)
\(\chi_{5881}(675,\cdot)\)
\(\chi_{5881}(676,\cdot)\)
\(\chi_{5881}(686,\cdot)\)
\(\chi_{5881}(709,\cdot)\)
\(\chi_{5881}(811,\cdot)\)
\(\chi_{5881}(896,\cdot)\)
...
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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 };
\(31\) → \(e\left(\frac{79}{588}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 5881 }(1081, a) \) |
\(1\) | \(1\) | \(e\left(\frac{127}{147}\right)\) | \(e\left(\frac{151}{294}\right)\) | \(e\left(\frac{107}{147}\right)\) | \(e\left(\frac{65}{294}\right)\) | \(e\left(\frac{37}{98}\right)\) | \(e\left(\frac{33}{98}\right)\) | \(e\left(\frac{29}{49}\right)\) | \(e\left(\frac{4}{147}\right)\) | \(e\left(\frac{25}{294}\right)\) | \(e\left(\frac{29}{196}\right)\) |
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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)
