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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(31115, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([63,93,86]))
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gp:[g,chi] = znchar(Mod(15284, 31115))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("31115.15284");
| Modulus: | \(31115\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(31115\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(126\) |
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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);
|
\(\chi_{31115}(479,\cdot)\)
\(\chi_{31115}(824,\cdot)\)
\(\chi_{31115}(1349,\cdot)\)
\(\chi_{31115}(1559,\cdot)\)
\(\chi_{31115}(4329,\cdot)\)
\(\chi_{31115}(4644,\cdot)\)
\(\chi_{31115}(4814,\cdot)\)
\(\chi_{31115}(5024,\cdot)\)
\(\chi_{31115}(5619,\cdot)\)
\(\chi_{31115}(6184,\cdot)\)
\(\chi_{31115}(7299,\cdot)\)
\(\chi_{31115}(8629,\cdot)\)
\(\chi_{31115}(9014,\cdot)\)
\(\chi_{31115}(10069,\cdot)\)
\(\chi_{31115}(10484,\cdot)\)
\(\chi_{31115}(10694,\cdot)\)
\(\chi_{31115}(12059,\cdot)\)
\(\chi_{31115}(12969,\cdot)\)
\(\chi_{31115}(14689,\cdot)\)
\(\chi_{31115}(14999,\cdot)\)
\(\chi_{31115}(15004,\cdot)\)
\(\chi_{31115}(15284,\cdot)\)
\(\chi_{31115}(15524,\cdot)\)
\(\chi_{31115}(18329,\cdot)\)
\(\chi_{31115}(19694,\cdot)\)
\(\chi_{31115}(20989,\cdot)\)
\(\chi_{31115}(21024,\cdot)\)
\(\chi_{31115}(21544,\cdot)\)
\(\chi_{31115}(23544,\cdot)\)
\(\chi_{31115}(24839,\cdot)\)
...
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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 };
\((12447,30481,6861)\) → \((-1,e\left(\frac{31}{42}\right),e\left(\frac{43}{63}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 31115 }(15284, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{95}{126}\right)\) | \(-1\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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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)
