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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13688, base_ring=CyclotomicField(406))
M = H._module
chi = DirichletCharacter(H, M([0,203,319,42]))
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gp:[g,chi] = znchar(Mod(5, 13688))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13688.5");
| Modulus: | \(13688\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(13688\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(406\) |
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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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\(\chi_{13688}(5,\cdot)\)
\(\chi_{13688}(125,\cdot)\)
\(\chi_{13688}(245,\cdot)\)
\(\chi_{13688}(341,\cdot)\)
\(\chi_{13688}(357,\cdot)\)
\(\chi_{13688}(381,\cdot)\)
\(\chi_{13688}(477,\cdot)\)
\(\chi_{13688}(557,\cdot)\)
\(\chi_{13688}(789,\cdot)\)
\(\chi_{13688}(845,\cdot)\)
\(\chi_{13688}(933,\cdot)\)
\(\chi_{13688}(1077,\cdot)\)
\(\chi_{13688}(1285,\cdot)\)
\(\chi_{13688}(1405,\cdot)\)
\(\chi_{13688}(1501,\cdot)\)
\(\chi_{13688}(1541,\cdot)\)
\(\chi_{13688}(1629,\cdot)\)
\(\chi_{13688}(1733,\cdot)\)
\(\chi_{13688}(1773,\cdot)\)
\(\chi_{13688}(2093,\cdot)\)
\(\chi_{13688}(2101,\cdot)\)
\(\chi_{13688}(2181,\cdot)\)
\(\chi_{13688}(2413,\cdot)\)
\(\chi_{13688}(2445,\cdot)\)
\(\chi_{13688}(2557,\cdot)\)
\(\chi_{13688}(2565,\cdot)\)
\(\chi_{13688}(2645,\cdot)\)
\(\chi_{13688}(2677,\cdot)\)
\(\chi_{13688}(2701,\cdot)\)
\(\chi_{13688}(2789,\cdot)\)
...
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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 };
| Field of values: |
$\Q(\zeta_{203})$ |
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sage:CyclotomicField(chi.multiplicative_order())
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gp:nfinit(polcyclo(charorder(g,chi)))
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magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 406 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((3423,6845,5193,10209)\) → \((1,-1,e\left(\frac{11}{14}\right),e\left(\frac{3}{29}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 13688 }(5, a) \) |
\(1\) | \(1\) | \(e\left(\frac{122}{203}\right)\) | \(e\left(\frac{165}{406}\right)\) | \(e\left(\frac{59}{203}\right)\) | \(e\left(\frac{41}{203}\right)\) | \(e\left(\frac{148}{203}\right)\) | \(e\left(\frac{121}{406}\right)\) | \(e\left(\frac{3}{406}\right)\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{102}{203}\right)\) | \(e\left(\frac{181}{203}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
