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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4729, base_ring=CyclotomicField(394))
M = H._module
chi = DirichletCharacter(H, M([386]))
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gp:[g,chi] = znchar(Mod(319, 4729))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4729.319");
| Modulus: | \(4729\) |
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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: | \(4729\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(197\) |
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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: | 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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\(\chi_{4729}(16,\cdot)\)
\(\chi_{4729}(24,\cdot)\)
\(\chi_{4729}(36,\cdot)\)
\(\chi_{4729}(54,\cdot)\)
\(\chi_{4729}(81,\cdot)\)
\(\chi_{4729}(250,\cdot)\)
\(\chi_{4729}(251,\cdot)\)
\(\chi_{4729}(253,\cdot)\)
\(\chi_{4729}(256,\cdot)\)
\(\chi_{4729}(295,\cdot)\)
\(\chi_{4729}(314,\cdot)\)
\(\chi_{4729}(319,\cdot)\)
\(\chi_{4729}(325,\cdot)\)
\(\chi_{4729}(375,\cdot)\)
\(\chi_{4729}(384,\cdot)\)
\(\chi_{4729}(471,\cdot)\)
\(\chi_{4729}(490,\cdot)\)
\(\chi_{4729}(497,\cdot)\)
\(\chi_{4729}(576,\cdot)\)
\(\chi_{4729}(637,\cdot)\)
\(\chi_{4729}(665,\cdot)\)
\(\chi_{4729}(682,\cdot)\)
\(\chi_{4729}(734,\cdot)\)
\(\chi_{4729}(735,\cdot)\)
\(\chi_{4729}(739,\cdot)\)
\(\chi_{4729}(746,\cdot)\)
\(\chi_{4729}(788,\cdot)\)
\(\chi_{4729}(821,\cdot)\)
\(\chi_{4729}(857,\cdot)\)
\(\chi_{4729}(864,\cdot)\)
...
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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: |
$\Q(\zeta_{197})$ |
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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));
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| Fixed field: |
Number field defined by a degree 197 polynomial (not computed) |
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sage:chi.fixed_field()
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\(17\) → \(e\left(\frac{193}{197}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4729 }(319, a) \) |
\(1\) | \(1\) | \(e\left(\frac{107}{197}\right)\) | \(e\left(\frac{108}{197}\right)\) | \(e\left(\frac{17}{197}\right)\) | \(e\left(\frac{169}{197}\right)\) | \(e\left(\frac{18}{197}\right)\) | \(e\left(\frac{143}{197}\right)\) | \(e\left(\frac{124}{197}\right)\) | \(e\left(\frac{19}{197}\right)\) | \(e\left(\frac{79}{197}\right)\) | \(e\left(\frac{97}{197}\right)\) |
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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)
