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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(18655, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([60,20,70,69]))
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gp:[g,chi] = znchar(Mod(10034, 18655))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("18655.10034");
| Modulus: | \(18655\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(18655\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(120\) |
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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: | odd |
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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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\(\chi_{18655}(19,\cdot)\)
\(\chi_{18655}(479,\cdot)\)
\(\chi_{18655}(1489,\cdot)\)
\(\chi_{18655}(1839,\cdot)\)
\(\chi_{18655}(2754,\cdot)\)
\(\chi_{18655}(3174,\cdot)\)
\(\chi_{18655}(3204,\cdot)\)
\(\chi_{18655}(3209,\cdot)\)
\(\chi_{18655}(3309,\cdot)\)
\(\chi_{18655}(5024,\cdot)\)
\(\chi_{18655}(5479,\cdot)\)
\(\chi_{18655}(6039,\cdot)\)
\(\chi_{18655}(7859,\cdot)\)
\(\chi_{18655}(8634,\cdot)\)
\(\chi_{18655}(9119,\cdot)\)
\(\chi_{18655}(10034,\cdot)\)
\(\chi_{18655}(10134,\cdot)\)
\(\chi_{18655}(10489,\cdot)\)
\(\chi_{18655}(10589,\cdot)\)
\(\chi_{18655}(12274,\cdot)\)
\(\chi_{18655}(12729,\cdot)\)
\(\chi_{18655}(12764,\cdot)\)
\(\chi_{18655}(14549,\cdot)\)
\(\chi_{18655}(14579,\cdot)\)
\(\chi_{18655}(14584,\cdot)\)
\(\chi_{18655}(15914,\cdot)\)
\(\chi_{18655}(17314,\cdot)\)
\(\chi_{18655}(17414,\cdot)\)
\(\chi_{18655}(17734,\cdot)\)
\(\chi_{18655}(17869,\cdot)\)
...
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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 };
\((3732,15991,11481,14561)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{7}{12}\right),e\left(\frac{23}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(16\) | \(17\) |
| \( \chi_{ 18655 }(10034, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(i\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{97}{120}\right)\) |
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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)
