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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(59168, base_ring=CyclotomicField(1032))
M = H._module
chi = DirichletCharacter(H, M([516,645,4]))
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gp:[g,chi] = znchar(Mod(2187, 59168))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("59168.2187");
| Modulus: | \(59168\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(59168\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1032\) |
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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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\(\chi_{59168}(123,\cdot)\)
\(\chi_{59168}(179,\cdot)\)
\(\chi_{59168}(467,\cdot)\)
\(\chi_{59168}(523,\cdot)\)
\(\chi_{59168}(811,\cdot)\)
\(\chi_{59168}(867,\cdot)\)
\(\chi_{59168}(1155,\cdot)\)
\(\chi_{59168}(1211,\cdot)\)
\(\chi_{59168}(1499,\cdot)\)
\(\chi_{59168}(1555,\cdot)\)
\(\chi_{59168}(1843,\cdot)\)
\(\chi_{59168}(1899,\cdot)\)
\(\chi_{59168}(2187,\cdot)\)
\(\chi_{59168}(2243,\cdot)\)
\(\chi_{59168}(2531,\cdot)\)
\(\chi_{59168}(2587,\cdot)\)
\(\chi_{59168}(2875,\cdot)\)
\(\chi_{59168}(2931,\cdot)\)
\(\chi_{59168}(3219,\cdot)\)
\(\chi_{59168}(3563,\cdot)\)
\(\chi_{59168}(3619,\cdot)\)
\(\chi_{59168}(3907,\cdot)\)
\(\chi_{59168}(3963,\cdot)\)
\(\chi_{59168}(4251,\cdot)\)
\(\chi_{59168}(4307,\cdot)\)
\(\chi_{59168}(4595,\cdot)\)
\(\chi_{59168}(4651,\cdot)\)
\(\chi_{59168}(4939,\cdot)\)
\(\chi_{59168}(4995,\cdot)\)
\(\chi_{59168}(5283,\cdot)\)
...
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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: |
$\Q(\zeta_{1032})$ |
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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 1032 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((25887,7397,25889)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{1}{258}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 59168 }(2187, a) \) |
\(1\) | \(1\) | \(e\left(\frac{391}{1032}\right)\) | \(e\left(\frac{385}{1032}\right)\) | \(e\left(\frac{361}{516}\right)\) | \(e\left(\frac{391}{516}\right)\) | \(e\left(\frac{95}{344}\right)\) | \(e\left(\frac{419}{1032}\right)\) | \(e\left(\frac{97}{129}\right)\) | \(e\left(\frac{23}{258}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{27}{344}\right)\) |
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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)
