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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10400, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([0,75,54,10]))
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gp:[g,chi] = znchar(Mod(1237, 10400))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10400.1237");
| Modulus: | \(10400\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(10400\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(120\) |
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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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\(\chi_{10400}(197,\cdot)\)
\(\chi_{10400}(813,\cdot)\)
\(\chi_{10400}(973,\cdot)\)
\(\chi_{10400}(1237,\cdot)\)
\(\chi_{10400}(1397,\cdot)\)
\(\chi_{10400}(1853,\cdot)\)
\(\chi_{10400}(2013,\cdot)\)
\(\chi_{10400}(2277,\cdot)\)
\(\chi_{10400}(2437,\cdot)\)
\(\chi_{10400}(3053,\cdot)\)
\(\chi_{10400}(3317,\cdot)\)
\(\chi_{10400}(3477,\cdot)\)
\(\chi_{10400}(3933,\cdot)\)
\(\chi_{10400}(4517,\cdot)\)
\(\chi_{10400}(4973,\cdot)\)
\(\chi_{10400}(5133,\cdot)\)
\(\chi_{10400}(5397,\cdot)\)
\(\chi_{10400}(6013,\cdot)\)
\(\chi_{10400}(6173,\cdot)\)
\(\chi_{10400}(6437,\cdot)\)
\(\chi_{10400}(6597,\cdot)\)
\(\chi_{10400}(7053,\cdot)\)
\(\chi_{10400}(7213,\cdot)\)
\(\chi_{10400}(7477,\cdot)\)
\(\chi_{10400}(7637,\cdot)\)
\(\chi_{10400}(8253,\cdot)\)
\(\chi_{10400}(8517,\cdot)\)
\(\chi_{10400}(8677,\cdot)\)
\(\chi_{10400}(9133,\cdot)\)
\(\chi_{10400}(9717,\cdot)\)
...
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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: |
$\Q(\zeta_{120})$ |
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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 120 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((1951,6501,4577,1601)\) → \((1,e\left(\frac{5}{8}\right),e\left(\frac{9}{20}\right),e\left(\frac{1}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 10400 }(1237, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{13}{120}\right)\) |
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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)
