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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3675, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([105,168,55]))
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gp:[g,chi] = znchar(Mod(1286, 3675))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3675.1286");
| Modulus: | \(3675\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3675\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(210\) |
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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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\(\chi_{3675}(131,\cdot)\)
\(\chi_{3675}(206,\cdot)\)
\(\chi_{3675}(236,\cdot)\)
\(\chi_{3675}(311,\cdot)\)
\(\chi_{3675}(341,\cdot)\)
\(\chi_{3675}(416,\cdot)\)
\(\chi_{3675}(446,\cdot)\)
\(\chi_{3675}(731,\cdot)\)
\(\chi_{3675}(761,\cdot)\)
\(\chi_{3675}(836,\cdot)\)
\(\chi_{3675}(866,\cdot)\)
\(\chi_{3675}(941,\cdot)\)
\(\chi_{3675}(971,\cdot)\)
\(\chi_{3675}(1046,\cdot)\)
\(\chi_{3675}(1181,\cdot)\)
\(\chi_{3675}(1286,\cdot)\)
\(\chi_{3675}(1361,\cdot)\)
\(\chi_{3675}(1466,\cdot)\)
\(\chi_{3675}(1496,\cdot)\)
\(\chi_{3675}(1571,\cdot)\)
\(\chi_{3675}(1706,\cdot)\)
\(\chi_{3675}(1781,\cdot)\)
\(\chi_{3675}(1811,\cdot)\)
\(\chi_{3675}(1886,\cdot)\)
\(\chi_{3675}(1916,\cdot)\)
\(\chi_{3675}(2021,\cdot)\)
\(\chi_{3675}(2096,\cdot)\)
\(\chi_{3675}(2231,\cdot)\)
\(\chi_{3675}(2306,\cdot)\)
\(\chi_{3675}(2336,\cdot)\)
...
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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: |
$\Q(\zeta_{105})$ |
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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 210 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((1226,1177,2551)\) → \((-1,e\left(\frac{4}{5}\right),e\left(\frac{11}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 3675 }(1286, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{210}\right)\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{163}{210}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{53}{210}\right)\) |
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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)
