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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([406,406,783,84]))
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gp:[g,chi] = znchar(Mod(595, 13688))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13688.595");
| 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: | 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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\(\chi_{13688}(3,\cdot)\)
\(\chi_{13688}(19,\cdot)\)
\(\chi_{13688}(27,\cdot)\)
\(\chi_{13688}(147,\cdot)\)
\(\chi_{13688}(163,\cdot)\)
\(\chi_{13688}(171,\cdot)\)
\(\chi_{13688}(243,\cdot)\)
\(\chi_{13688}(251,\cdot)\)
\(\chi_{13688}(363,\cdot)\)
\(\chi_{13688}(379,\cdot)\)
\(\chi_{13688}(395,\cdot)\)
\(\chi_{13688}(403,\cdot)\)
\(\chi_{13688}(475,\cdot)\)
\(\chi_{13688}(491,\cdot)\)
\(\chi_{13688}(507,\cdot)\)
\(\chi_{13688}(595,\cdot)\)
\(\chi_{13688}(611,\cdot)\)
\(\chi_{13688}(619,\cdot)\)
\(\chi_{13688}(635,\cdot)\)
\(\chi_{13688}(675,\cdot)\)
\(\chi_{13688}(715,\cdot)\)
\(\chi_{13688}(723,\cdot)\)
\(\chi_{13688}(843,\cdot)\)
\(\chi_{13688}(851,\cdot)\)
\(\chi_{13688}(867,\cdot)\)
\(\chi_{13688}(907,\cdot)\)
\(\chi_{13688}(931,\cdot)\)
\(\chi_{13688}(947,\cdot)\)
\(\chi_{13688}(971,\cdot)\)
\(\chi_{13688}(1083,\cdot)\)
...
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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()
|
\((3423,6845,5193,10209)\) → \((-1,-1,e\left(\frac{27}{28}\right),e\left(\frac{3}{29}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 13688 }(595, a) \) |
\(1\) | \(1\) | \(e\left(\frac{807}{812}\right)\) | \(e\left(\frac{68}{203}\right)\) | \(e\left(\frac{379}{406}\right)\) | \(e\left(\frac{401}{406}\right)\) | \(e\left(\frac{563}{812}\right)\) | \(e\left(\frac{104}{203}\right)\) | \(e\left(\frac{267}{812}\right)\) | \(e\left(\frac{45}{116}\right)\) | \(e\left(\frac{495}{812}\right)\) | \(e\left(\frac{753}{812}\right)\) |
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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)
