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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11200, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([60,15,18,100]))
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gp:[g,chi] = znchar(Mod(4583, 11200))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11200.4583");
| Modulus: | \(11200\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5600\) |
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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: | no, induced from \(\chi_{5600}(1083,\cdot)\) |
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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: | no |
| 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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\(\chi_{11200}(87,\cdot)\)
\(\chi_{11200}(103,\cdot)\)
\(\chi_{11200}(423,\cdot)\)
\(\chi_{11200}(887,\cdot)\)
\(\chi_{11200}(1223,\cdot)\)
\(\chi_{11200}(2327,\cdot)\)
\(\chi_{11200}(2663,\cdot)\)
\(\chi_{11200}(3127,\cdot)\)
\(\chi_{11200}(3447,\cdot)\)
\(\chi_{11200}(3463,\cdot)\)
\(\chi_{11200}(3783,\cdot)\)
\(\chi_{11200}(4247,\cdot)\)
\(\chi_{11200}(4567,\cdot)\)
\(\chi_{11200}(4583,\cdot)\)
\(\chi_{11200}(4903,\cdot)\)
\(\chi_{11200}(5367,\cdot)\)
\(\chi_{11200}(5687,\cdot)\)
\(\chi_{11200}(5703,\cdot)\)
\(\chi_{11200}(6023,\cdot)\)
\(\chi_{11200}(6487,\cdot)\)
\(\chi_{11200}(6823,\cdot)\)
\(\chi_{11200}(7927,\cdot)\)
\(\chi_{11200}(8263,\cdot)\)
\(\chi_{11200}(8727,\cdot)\)
\(\chi_{11200}(9047,\cdot)\)
\(\chi_{11200}(9063,\cdot)\)
\(\chi_{11200}(9383,\cdot)\)
\(\chi_{11200}(9847,\cdot)\)
\(\chi_{11200}(10167,\cdot)\)
\(\chi_{11200}(10183,\cdot)\)
...
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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()
|
\((5951,10501,5377,6401)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{3}{20}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 11200 }(4583, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{8}{15}\right)\) |
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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)
