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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(836352, base_ring=CyclotomicField(6336))
M = H._module
chi = DirichletCharacter(H, M([3168,5643,4928,5184]))
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gp:[g,chi] = znchar(Mod(859, 836352))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("836352.859");
| Modulus: | \(836352\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(836352\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(6336\) |
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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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\(\chi_{836352}(67,\cdot)\)
\(\chi_{836352}(331,\cdot)\)
\(\chi_{836352}(859,\cdot)\)
\(\chi_{836352}(1123,\cdot)\)
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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: |
$\Q(\zeta_{6336})$ |
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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 6336 polynomial (not computed) |
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sage:chi.fixed_field()
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\((137215,561925,123905,781057)\) → \((-1,e\left(\frac{57}{64}\right),e\left(\frac{7}{9}\right),e\left(\frac{9}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 836352 }(859, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{2059}{6336}\right)\) | \(e\left(\frac{1831}{3168}\right)\) | \(e\left(\frac{4549}{6336}\right)\) | \(e\left(\frac{367}{528}\right)\) | \(e\left(\frac{479}{2112}\right)\) | \(e\left(\frac{2525}{3168}\right)\) | \(e\left(\frac{2059}{3168}\right)\) | \(e\left(\frac{1481}{6336}\right)\) | \(e\left(\frac{431}{792}\right)\) | \(e\left(\frac{1907}{2112}\right)\) |
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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)
