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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(349830, base_ring=CyclotomicField(1716))
M = H._module
chi = DirichletCharacter(H, M([0,429,242,312]))
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gp:[g,chi] = znchar(Mod(21187, 349830))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("349830.21187");
| Modulus: | \(349830\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(19435\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1716\) |
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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: | no, induced from \(\chi_{19435}(1752,\cdot)\) |
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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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\(\chi_{349830}(127,\cdot)\)
\(\chi_{349830}(1297,\cdot)\)
\(\chi_{349830}(1603,\cdot)\)
\(\chi_{349830}(2233,\cdot)\)
\(\chi_{349830}(2467,\cdot)\)
\(\chi_{349830}(2773,\cdot)\)
\(\chi_{349830}(3637,\cdot)\)
\(\chi_{349830}(5743,\cdot)\)
\(\chi_{349830}(6283,\cdot)\)
\(\chi_{349830}(6517,\cdot)\)
\(\chi_{349830}(6913,\cdot)\)
\(\chi_{349830}(8857,\cdot)\)
\(\chi_{349830}(9793,\cdot)\)
\(\chi_{349830}(10423,\cdot)\)
\(\chi_{349830}(10657,\cdot)\)
\(\chi_{349830}(12133,\cdot)\)
\(\chi_{349830}(12367,\cdot)\)
\(\chi_{349830}(12997,\cdot)\)
\(\chi_{349830}(13303,\cdot)\)
\(\chi_{349830}(13537,\cdot)\)
\(\chi_{349830}(13933,\cdot)\)
\(\chi_{349830}(14473,\cdot)\)
\(\chi_{349830}(15643,\cdot)\)
\(\chi_{349830}(16273,\cdot)\)
\(\chi_{349830}(16507,\cdot)\)
\(\chi_{349830}(17443,\cdot)\)
\(\chi_{349830}(17677,\cdot)\)
\(\chi_{349830}(19783,\cdot)\)
\(\chi_{349830}(20557,\cdot)\)
\(\chi_{349830}(21187,\cdot)\)
...
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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: |
$\Q(\zeta_{1716})$ |
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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 1716 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((310961,69967,341551,258571)\) → \((1,i,e\left(\frac{11}{78}\right),e\left(\frac{2}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 349830 }(21187, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1363}{1716}\right)\) | \(e\left(\frac{139}{858}\right)\) | \(e\left(\frac{193}{1716}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{355}{858}\right)\) | \(e\left(\frac{15}{286}\right)\) | \(e\left(\frac{623}{1716}\right)\) | \(e\left(\frac{145}{858}\right)\) | \(e\left(\frac{1483}{1716}\right)\) | \(e\left(\frac{7}{52}\right)\) |
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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)
