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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([52,29]))
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gp:[g,chi] = znchar(Mod(331, 1183))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1183.331");
| 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);
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| 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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\(\chi_{1183}(11,\cdot)\)
\(\chi_{1183}(58,\cdot)\)
\(\chi_{1183}(67,\cdot)\)
\(\chi_{1183}(72,\cdot)\)
\(\chi_{1183}(102,\cdot)\)
\(\chi_{1183}(149,\cdot)\)
\(\chi_{1183}(158,\cdot)\)
\(\chi_{1183}(163,\cdot)\)
\(\chi_{1183}(193,\cdot)\)
\(\chi_{1183}(240,\cdot)\)
\(\chi_{1183}(254,\cdot)\)
\(\chi_{1183}(284,\cdot)\)
\(\chi_{1183}(331,\cdot)\)
\(\chi_{1183}(340,\cdot)\)
\(\chi_{1183}(345,\cdot)\)
\(\chi_{1183}(375,\cdot)\)
\(\chi_{1183}(422,\cdot)\)
\(\chi_{1183}(431,\cdot)\)
\(\chi_{1183}(436,\cdot)\)
\(\chi_{1183}(466,\cdot)\)
\(\chi_{1183}(513,\cdot)\)
\(\chi_{1183}(522,\cdot)\)
\(\chi_{1183}(527,\cdot)\)
\(\chi_{1183}(557,\cdot)\)
\(\chi_{1183}(604,\cdot)\)
\(\chi_{1183}(613,\cdot)\)
\(\chi_{1183}(618,\cdot)\)
\(\chi_{1183}(648,\cdot)\)
\(\chi_{1183}(704,\cdot)\)
\(\chi_{1183}(709,\cdot)\)
...
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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()
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\((339,1016)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{29}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 1183 }(331, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{133}{156}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{53}{156}\right)\) | \(e\left(\frac{37}{156}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{7}{78}\right)\) |
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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)
