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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1063, base_ring=CyclotomicField(1062))
M = H._module
chi = DirichletCharacter(H, M([709]))
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gp:[g,chi] = znchar(Mod(31, 1063))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1063.31");
| Modulus: | \(1063\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1063\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1062\) |
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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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\(\chi_{1063}(3,\cdot)\)
\(\chi_{1063}(6,\cdot)\)
\(\chi_{1063}(10,\cdot)\)
\(\chi_{1063}(20,\cdot)\)
\(\chi_{1063}(21,\cdot)\)
\(\chi_{1063}(24,\cdot)\)
\(\chi_{1063}(29,\cdot)\)
\(\chi_{1063}(31,\cdot)\)
\(\chi_{1063}(33,\cdot)\)
\(\chi_{1063}(35,\cdot)\)
\(\chi_{1063}(39,\cdot)\)
\(\chi_{1063}(45,\cdot)\)
\(\chi_{1063}(48,\cdot)\)
\(\chi_{1063}(51,\cdot)\)
\(\chi_{1063}(53,\cdot)\)
\(\chi_{1063}(54,\cdot)\)
\(\chi_{1063}(55,\cdot)\)
\(\chi_{1063}(58,\cdot)\)
\(\chi_{1063}(61,\cdot)\)
\(\chi_{1063}(62,\cdot)\)
\(\chi_{1063}(69,\cdot)\)
\(\chi_{1063}(70,\cdot)\)
\(\chi_{1063}(75,\cdot)\)
\(\chi_{1063}(78,\cdot)\)
\(\chi_{1063}(79,\cdot)\)
\(\chi_{1063}(80,\cdot)\)
\(\chi_{1063}(82,\cdot)\)
\(\chi_{1063}(84,\cdot)\)
\(\chi_{1063}(86,\cdot)\)
\(\chi_{1063}(95,\cdot)\)
...
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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: |
$\Q(\zeta_{531})$ |
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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 1062 polynomial (not computed) |
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sage:chi.fixed_field()
|
\(3\) → \(e\left(\frac{709}{1062}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1063 }(31, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{347}{531}\right)\) | \(e\left(\frac{709}{1062}\right)\) | \(e\left(\frac{163}{531}\right)\) | \(e\left(\frac{313}{354}\right)\) | \(e\left(\frac{341}{1062}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{170}{177}\right)\) | \(e\left(\frac{178}{531}\right)\) | \(e\left(\frac{571}{1062}\right)\) | \(e\left(\frac{182}{531}\right)\) |
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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)
