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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1063, base_ring=CyclotomicField(354))
M = H._module
chi = DirichletCharacter(H, M([85]))
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gp:[g,chi] = znchar(Mod(107, 1063))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1063.107");
| Modulus: | \(1063\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
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| 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: | \(354\) |
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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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\(\chi_{1063}(5,\cdot)\)
\(\chi_{1063}(27,\cdot)\)
\(\chi_{1063}(40,\cdot)\)
\(\chi_{1063}(41,\cdot)\)
\(\chi_{1063}(42,\cdot)\)
\(\chi_{1063}(43,\cdot)\)
\(\chi_{1063}(57,\cdot)\)
\(\chi_{1063}(59,\cdot)\)
\(\chi_{1063}(65,\cdot)\)
\(\chi_{1063}(66,\cdot)\)
\(\chi_{1063}(85,\cdot)\)
\(\chi_{1063}(96,\cdot)\)
\(\chi_{1063}(106,\cdot)\)
\(\chi_{1063}(107,\cdot)\)
\(\chi_{1063}(113,\cdot)\)
\(\chi_{1063}(116,\cdot)\)
\(\chi_{1063}(124,\cdot)\)
\(\chi_{1063}(140,\cdot)\)
\(\chi_{1063}(141,\cdot)\)
\(\chi_{1063}(147,\cdot)\)
\(\chi_{1063}(156,\cdot)\)
\(\chi_{1063}(185,\cdot)\)
\(\chi_{1063}(190,\cdot)\)
\(\chi_{1063}(216,\cdot)\)
\(\chi_{1063}(244,\cdot)\)
\(\chi_{1063}(254,\cdot)\)
\(\chi_{1063}(261,\cdot)\)
\(\chi_{1063}(271,\cdot)\)
\(\chi_{1063}(277,\cdot)\)
\(\chi_{1063}(283,\cdot)\)
...
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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: |
$\Q(\zeta_{177})$ |
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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 354 polynomial (not computed) |
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sage:chi.fixed_field()
|
\(3\) → \(e\left(\frac{85}{354}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1063 }(107, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{113}{177}\right)\) | \(e\left(\frac{85}{354}\right)\) | \(e\left(\frac{49}{177}\right)\) | \(e\left(\frac{55}{118}\right)\) | \(e\left(\frac{311}{354}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{54}{59}\right)\) | \(e\left(\frac{85}{177}\right)\) | \(e\left(\frac{37}{354}\right)\) | \(e\left(\frac{71}{177}\right)\) |
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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)
