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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8021, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([22,75]))
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gp:[g,chi] = znchar(Mod(290, 8021))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8021.290");
| Modulus: | \(8021\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8021\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(132\) |
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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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\(\chi_{8021}(69,\cdot)\)
\(\chi_{8021}(290,\cdot)\)
\(\chi_{8021}(329,\cdot)\)
\(\chi_{8021}(465,\cdot)\)
\(\chi_{8021}(517,\cdot)\)
\(\chi_{8021}(602,\cdot)\)
\(\chi_{8021}(686,\cdot)\)
\(\chi_{8021}(907,\cdot)\)
\(\chi_{8021}(946,\cdot)\)
\(\chi_{8021}(1219,\cdot)\)
\(\chi_{8021}(1499,\cdot)\)
\(\chi_{8021}(1694,\cdot)\)
\(\chi_{8021}(2116,\cdot)\)
\(\chi_{8021}(2311,\cdot)\)
\(\chi_{8021}(2929,\cdot)\)
\(\chi_{8021}(3241,\cdot)\)
\(\chi_{8021}(3546,\cdot)\)
\(\chi_{8021}(3858,\cdot)\)
\(\chi_{8021}(4476,\cdot)\)
\(\chi_{8021}(4671,\cdot)\)
\(\chi_{8021}(5093,\cdot)\)
\(\chi_{8021}(5288,\cdot)\)
\(\chi_{8021}(5568,\cdot)\)
\(\chi_{8021}(5841,\cdot)\)
\(\chi_{8021}(5880,\cdot)\)
\(\chi_{8021}(6101,\cdot)\)
\(\chi_{8021}(6185,\cdot)\)
\(\chi_{8021}(6270,\cdot)\)
\(\chi_{8021}(6322,\cdot)\)
\(\chi_{8021}(6458,\cdot)\)
...
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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 };
\((6788,2471)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{25}{44}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8021 }(290, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{31}{33}\right)\) |
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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)
