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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1359, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([25,26]))
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gp:[g,chi] = znchar(Mod(47, 1359))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1359.47");
| Modulus: | \(1359\) |
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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: | \(1359\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(150\) |
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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;
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| 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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\(\chi_{1359}(5,\cdot)\)
\(\chi_{1359}(47,\cdot)\)
\(\chi_{1359}(74,\cdot)\)
\(\chi_{1359}(95,\cdot)\)
\(\chi_{1359}(137,\cdot)\)
\(\chi_{1359}(173,\cdot)\)
\(\chi_{1359}(200,\cdot)\)
\(\chi_{1359}(239,\cdot)\)
\(\chi_{1359}(248,\cdot)\)
\(\chi_{1359}(272,\cdot)\)
\(\chi_{1359}(290,\cdot)\)
\(\chi_{1359}(320,\cdot)\)
\(\chi_{1359}(344,\cdot)\)
\(\chi_{1359}(347,\cdot)\)
\(\chi_{1359}(401,\cdot)\)
\(\chi_{1359}(446,\cdot)\)
\(\chi_{1359}(464,\cdot)\)
\(\chi_{1359}(515,\cdot)\)
\(\chi_{1359}(533,\cdot)\)
\(\chi_{1359}(569,\cdot)\)
\(\chi_{1359}(614,\cdot)\)
\(\chi_{1359}(635,\cdot)\)
\(\chi_{1359}(644,\cdot)\)
\(\chi_{1359}(659,\cdot)\)
\(\chi_{1359}(662,\cdot)\)
\(\chi_{1359}(707,\cdot)\)
\(\chi_{1359}(794,\cdot)\)
\(\chi_{1359}(824,\cdot)\)
\(\chi_{1359}(893,\cdot)\)
\(\chi_{1359}(923,\cdot)\)
...
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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: |
$\Q(\zeta_{75})$ |
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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 150 polynomial (not computed) |
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sage:chi.fixed_field()
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\((605,1063)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{13}{75}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1359 }(47, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{37}{150}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{1}{5}\right)\) |
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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)
