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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1183, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([52,9]))
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gp:[g,chi] = znchar(Mod(681, 1183))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1183.681");
| Modulus: | \(1183\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1183\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(156\) |
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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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\(\chi_{1183}(18,\cdot)\)
\(\chi_{1183}(44,\cdot)\)
\(\chi_{1183}(60,\cdot)\)
\(\chi_{1183}(86,\cdot)\)
\(\chi_{1183}(109,\cdot)\)
\(\chi_{1183}(135,\cdot)\)
\(\chi_{1183}(151,\cdot)\)
\(\chi_{1183}(177,\cdot)\)
\(\chi_{1183}(200,\cdot)\)
\(\chi_{1183}(226,\cdot)\)
\(\chi_{1183}(242,\cdot)\)
\(\chi_{1183}(291,\cdot)\)
\(\chi_{1183}(317,\cdot)\)
\(\chi_{1183}(333,\cdot)\)
\(\chi_{1183}(359,\cdot)\)
\(\chi_{1183}(382,\cdot)\)
\(\chi_{1183}(424,\cdot)\)
\(\chi_{1183}(450,\cdot)\)
\(\chi_{1183}(473,\cdot)\)
\(\chi_{1183}(499,\cdot)\)
\(\chi_{1183}(515,\cdot)\)
\(\chi_{1183}(541,\cdot)\)
\(\chi_{1183}(564,\cdot)\)
\(\chi_{1183}(590,\cdot)\)
\(\chi_{1183}(632,\cdot)\)
\(\chi_{1183}(655,\cdot)\)
\(\chi_{1183}(681,\cdot)\)
\(\chi_{1183}(697,\cdot)\)
\(\chi_{1183}(723,\cdot)\)
\(\chi_{1183}(772,\cdot)\)
...
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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: |
$\Q(\zeta_{156})$ |
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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 156 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((339,1016)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{3}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 1183 }(681, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{113}{156}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{29}{156}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{43}{156}\right)\) | \(e\left(\frac{73}{78}\right)\) |
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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)
