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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13688, base_ring=CyclotomicField(812))
M = H._module
chi = DirichletCharacter(H, M([406,406,667,770]))
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gp:[g,chi] = znchar(Mod(155, 13688))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13688.155");
| 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: | \(812\) |
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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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\(\chi_{13688}(11,\cdot)\)
\(\chi_{13688}(43,\cdot)\)
\(\chi_{13688}(131,\cdot)\)
\(\chi_{13688}(155,\cdot)\)
\(\chi_{13688}(195,\cdot)\)
\(\chi_{13688}(211,\cdot)\)
\(\chi_{13688}(259,\cdot)\)
\(\chi_{13688}(275,\cdot)\)
\(\chi_{13688}(387,\cdot)\)
\(\chi_{13688}(427,\cdot)\)
\(\chi_{13688}(443,\cdot)\)
\(\chi_{13688}(467,\cdot)\)
\(\chi_{13688}(483,\cdot)\)
\(\chi_{13688}(627,\cdot)\)
\(\chi_{13688}(659,\cdot)\)
\(\chi_{13688}(699,\cdot)\)
\(\chi_{13688}(739,\cdot)\)
\(\chi_{13688}(859,\cdot)\)
\(\chi_{13688}(891,\cdot)\)
\(\chi_{13688}(939,\cdot)\)
\(\chi_{13688}(955,\cdot)\)
\(\chi_{13688}(1059,\cdot)\)
\(\chi_{13688}(1075,\cdot)\)
\(\chi_{13688}(1099,\cdot)\)
\(\chi_{13688}(1123,\cdot)\)
\(\chi_{13688}(1139,\cdot)\)
\(\chi_{13688}(1163,\cdot)\)
\(\chi_{13688}(1171,\cdot)\)
\(\chi_{13688}(1203,\cdot)\)
\(\chi_{13688}(1291,\cdot)\)
...
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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: |
$\Q(\zeta_{812})$ |
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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 812 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((3423,6845,5193,10209)\) → \((-1,-1,e\left(\frac{23}{28}\right),e\left(\frac{55}{58}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 13688 }(155, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{423}{812}\right)\) | \(e\left(\frac{53}{203}\right)\) | \(e\left(\frac{173}{406}\right)\) | \(e\left(\frac{17}{406}\right)\) | \(e\left(\frac{197}{812}\right)\) | \(e\left(\frac{389}{406}\right)\) | \(e\left(\frac{635}{812}\right)\) | \(e\left(\frac{21}{116}\right)\) | \(e\left(\frac{347}{812}\right)\) | \(e\left(\frac{769}{812}\right)\) |
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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)
