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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38663, base_ring=CyclotomicField(410))
M = H._module
chi = DirichletCharacter(H, M([205,118]))
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gp:[g,chi] = znchar(Mod(1103, 38663))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("38663.1103");
| Modulus: | \(38663\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(38663\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(410\) |
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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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\(\chi_{38663}(160,\cdot)\)
\(\chi_{38663}(344,\cdot)\)
\(\chi_{38663}(551,\cdot)\)
\(\chi_{38663}(666,\cdot)\)
\(\chi_{38663}(1103,\cdot)\)
\(\chi_{38663}(1287,\cdot)\)
\(\chi_{38663}(1494,\cdot)\)
\(\chi_{38663}(1609,\cdot)\)
\(\chi_{38663}(2046,\cdot)\)
\(\chi_{38663}(2230,\cdot)\)
\(\chi_{38663}(2437,\cdot)\)
\(\chi_{38663}(2552,\cdot)\)
\(\chi_{38663}(2989,\cdot)\)
\(\chi_{38663}(3173,\cdot)\)
\(\chi_{38663}(3380,\cdot)\)
\(\chi_{38663}(3495,\cdot)\)
\(\chi_{38663}(3932,\cdot)\)
\(\chi_{38663}(4116,\cdot)\)
\(\chi_{38663}(4323,\cdot)\)
\(\chi_{38663}(4438,\cdot)\)
\(\chi_{38663}(4875,\cdot)\)
\(\chi_{38663}(5059,\cdot)\)
\(\chi_{38663}(5266,\cdot)\)
\(\chi_{38663}(5381,\cdot)\)
\(\chi_{38663}(5818,\cdot)\)
\(\chi_{38663}(6002,\cdot)\)
\(\chi_{38663}(6209,\cdot)\)
\(\chi_{38663}(6324,\cdot)\)
\(\chi_{38663}(6761,\cdot)\)
\(\chi_{38663}(6945,\cdot)\)
...
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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 };
\((3363,15135)\) → \((-1,e\left(\frac{59}{205}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 38663 }(1103, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{179}{205}\right)\) | \(e\left(\frac{17}{41}\right)\) | \(e\left(\frac{153}{205}\right)\) | \(e\left(\frac{291}{410}\right)\) | \(e\left(\frac{59}{205}\right)\) | \(e\left(\frac{157}{410}\right)\) | \(e\left(\frac{127}{205}\right)\) | \(e\left(\frac{34}{41}\right)\) | \(e\left(\frac{239}{410}\right)\) | \(e\left(\frac{229}{410}\right)\) |
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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)
