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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([26,93]))
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gp:[g,chi] = znchar(Mod(1032, 1183))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1183.1032");
| 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: | 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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\(\chi_{1183}(5,\cdot)\)
\(\chi_{1183}(31,\cdot)\)
\(\chi_{1183}(47,\cdot)\)
\(\chi_{1183}(73,\cdot)\)
\(\chi_{1183}(96,\cdot)\)
\(\chi_{1183}(122,\cdot)\)
\(\chi_{1183}(138,\cdot)\)
\(\chi_{1183}(164,\cdot)\)
\(\chi_{1183}(187,\cdot)\)
\(\chi_{1183}(213,\cdot)\)
\(\chi_{1183}(229,\cdot)\)
\(\chi_{1183}(255,\cdot)\)
\(\chi_{1183}(278,\cdot)\)
\(\chi_{1183}(304,\cdot)\)
\(\chi_{1183}(320,\cdot)\)
\(\chi_{1183}(346,\cdot)\)
\(\chi_{1183}(369,\cdot)\)
\(\chi_{1183}(395,\cdot)\)
\(\chi_{1183}(411,\cdot)\)
\(\chi_{1183}(460,\cdot)\)
\(\chi_{1183}(486,\cdot)\)
\(\chi_{1183}(502,\cdot)\)
\(\chi_{1183}(528,\cdot)\)
\(\chi_{1183}(551,\cdot)\)
\(\chi_{1183}(593,\cdot)\)
\(\chi_{1183}(619,\cdot)\)
\(\chi_{1183}(642,\cdot)\)
\(\chi_{1183}(668,\cdot)\)
\(\chi_{1183}(684,\cdot)\)
\(\chi_{1183}(710,\cdot)\)
...
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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()
|
\((339,1016)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{31}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 1183 }(1032, a) \) |
\(1\) | \(1\) | \(e\left(\frac{145}{156}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{31}{156}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{11}{156}\right)\) | \(e\left(\frac{37}{39}\right)\) |
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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)
