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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3661, base_ring=CyclotomicField(174))
M = H._module
chi = DirichletCharacter(H, M([29,59]))
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gp:[g,chi] = znchar(Mod(1081, 3661))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3661.1081");
| Modulus: | \(3661\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3661\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(174\) |
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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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\(\chi_{3661}(26,\cdot)\)
\(\chi_{3661}(38,\cdot)\)
\(\chi_{3661}(101,\cdot)\)
\(\chi_{3661}(145,\cdot)\)
\(\chi_{3661}(192,\cdot)\)
\(\chi_{3661}(220,\cdot)\)
\(\chi_{3661}(409,\cdot)\)
\(\chi_{3661}(481,\cdot)\)
\(\chi_{3661}(537,\cdot)\)
\(\chi_{3661}(649,\cdot)\)
\(\chi_{3661}(677,\cdot)\)
\(\chi_{3661}(682,\cdot)\)
\(\chi_{3661}(703,\cdot)\)
\(\chi_{3661}(782,\cdot)\)
\(\chi_{3661}(850,\cdot)\)
\(\chi_{3661}(852,\cdot)\)
\(\chi_{3661}(934,\cdot)\)
\(\chi_{3661}(941,\cdot)\)
\(\chi_{3661}(1081,\cdot)\)
\(\chi_{3661}(1118,\cdot)\)
\(\chi_{3661}(1125,\cdot)\)
\(\chi_{3661}(1361,\cdot)\)
\(\chi_{3661}(1382,\cdot)\)
\(\chi_{3661}(1431,\cdot)\)
\(\chi_{3661}(1545,\cdot)\)
\(\chi_{3661}(1620,\cdot)\)
\(\chi_{3661}(1634,\cdot)\)
\(\chi_{3661}(1657,\cdot)\)
\(\chi_{3661}(1692,\cdot)\)
\(\chi_{3661}(1909,\cdot)\)
...
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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 };
\((3139,2094)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{59}{174}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 3661 }(1081, a) \) |
\(1\) | \(1\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{76}{87}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{95}{174}\right)\) | \(e\left(\frac{1}{58}\right)\) | \(e\left(\frac{65}{87}\right)\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{19}{87}\right)\) | \(e\left(\frac{19}{87}\right)\) |
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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)
