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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([25,48]))
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gp:[g,chi] = znchar(Mod(974, 1359))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1359.974");
| Modulus: | \(1359\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| 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;
|
| 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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\(\chi_{1359}(20,\cdot)\)
\(\chi_{1359}(29,\cdot)\)
\(\chi_{1359}(50,\cdot)\)
\(\chi_{1359}(68,\cdot)\)
\(\chi_{1359}(86,\cdot)\)
\(\chi_{1359}(110,\cdot)\)
\(\chi_{1359}(245,\cdot)\)
\(\chi_{1359}(275,\cdot)\)
\(\chi_{1359}(299,\cdot)\)
\(\chi_{1359}(311,\cdot)\)
\(\chi_{1359}(374,\cdot)\)
\(\chi_{1359}(380,\cdot)\)
\(\chi_{1359}(383,\cdot)\)
\(\chi_{1359}(425,\cdot)\)
\(\chi_{1359}(473,\cdot)\)
\(\chi_{1359}(482,\cdot)\)
\(\chi_{1359}(497,\cdot)\)
\(\chi_{1359}(551,\cdot)\)
\(\chi_{1359}(563,\cdot)\)
\(\chi_{1359}(578,\cdot)\)
\(\chi_{1359}(695,\cdot)\)
\(\chi_{1359}(698,\cdot)\)
\(\chi_{1359}(731,\cdot)\)
\(\chi_{1359}(752,\cdot)\)
\(\chi_{1359}(833,\cdot)\)
\(\chi_{1359}(839,\cdot)\)
\(\chi_{1359}(878,\cdot)\)
\(\chi_{1359}(950,\cdot)\)
\(\chi_{1359}(956,\cdot)\)
\(\chi_{1359}(974,\cdot)\)
...
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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()
|
\((605,1063)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{8}{25}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1359 }(974, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{101}{150}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{37}{150}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{101}{150}\right)\) | \(e\left(\frac{4}{15}\right)\) |
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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)
