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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(22264, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([55,55,27,100]))
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gp:[g,chi] = znchar(Mod(14571, 22264))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("22264.14571");
| Modulus: | \(22264\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(22264\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(110\) |
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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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\(\chi_{22264}(491,\cdot)\)
\(\chi_{22264}(1267,\cdot)\)
\(\chi_{22264}(1899,\cdot)\)
\(\chi_{22264}(2923,\cdot)\)
\(\chi_{22264}(3163,\cdot)\)
\(\chi_{22264}(3571,\cdot)\)
\(\chi_{22264}(3659,\cdot)\)
\(\chi_{22264}(5155,\cdot)\)
\(\chi_{22264}(5651,\cdot)\)
\(\chi_{22264}(6035,\cdot)\)
\(\chi_{22264}(6683,\cdot)\)
\(\chi_{22264}(6771,\cdot)\)
\(\chi_{22264}(6811,\cdot)\)
\(\chi_{22264}(7355,\cdot)\)
\(\chi_{22264}(7387,\cdot)\)
\(\chi_{22264}(8907,\cdot)\)
\(\chi_{22264}(9963,\cdot)\)
\(\chi_{22264}(10115,\cdot)\)
\(\chi_{22264}(11035,\cdot)\)
\(\chi_{22264}(11171,\cdot)\)
\(\chi_{22264}(11283,\cdot)\)
\(\chi_{22264}(11963,\cdot)\)
\(\chi_{22264}(12403,\cdot)\)
\(\chi_{22264}(12459,\cdot)\)
\(\chi_{22264}(12515,\cdot)\)
\(\chi_{22264}(12723,\cdot)\)
\(\chi_{22264}(14203,\cdot)\)
\(\chi_{22264}(14571,\cdot)\)
\(\chi_{22264}(15347,\cdot)\)
\(\chi_{22264}(15699,\cdot)\)
...
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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 };
\((5567,11133,13433,1937)\) → \((-1,-1,e\left(\frac{27}{110}\right),e\left(\frac{10}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(25\) |
| \( \chi_{ 22264 }(14571, a) \) |
\(1\) | \(1\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{8}{55}\right)\) |
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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)
