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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5600, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([60,105,84,20]))
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gp:[g,chi] = znchar(Mod(1459, 5600))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5600.1459");
| Modulus: | \(5600\) |
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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;
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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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\(\chi_{5600}(19,\cdot)\)
\(\chi_{5600}(59,\cdot)\)
\(\chi_{5600}(339,\cdot)\)
\(\chi_{5600}(579,\cdot)\)
\(\chi_{5600}(619,\cdot)\)
\(\chi_{5600}(859,\cdot)\)
\(\chi_{5600}(1139,\cdot)\)
\(\chi_{5600}(1179,\cdot)\)
\(\chi_{5600}(1419,\cdot)\)
\(\chi_{5600}(1459,\cdot)\)
\(\chi_{5600}(1739,\cdot)\)
\(\chi_{5600}(1979,\cdot)\)
\(\chi_{5600}(2019,\cdot)\)
\(\chi_{5600}(2259,\cdot)\)
\(\chi_{5600}(2539,\cdot)\)
\(\chi_{5600}(2579,\cdot)\)
\(\chi_{5600}(2819,\cdot)\)
\(\chi_{5600}(2859,\cdot)\)
\(\chi_{5600}(3139,\cdot)\)
\(\chi_{5600}(3379,\cdot)\)
\(\chi_{5600}(3419,\cdot)\)
\(\chi_{5600}(3659,\cdot)\)
\(\chi_{5600}(3939,\cdot)\)
\(\chi_{5600}(3979,\cdot)\)
\(\chi_{5600}(4219,\cdot)\)
\(\chi_{5600}(4259,\cdot)\)
\(\chi_{5600}(4539,\cdot)\)
\(\chi_{5600}(4779,\cdot)\)
\(\chi_{5600}(4819,\cdot)\)
\(\chi_{5600}(5059,\cdot)\)
...
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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));
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| Fixed field: |
Number field defined by a degree 120 polynomial (not computed) |
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sage:chi.fixed_field()
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\((351,4901,5377,801)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{7}{10}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 5600 }(1459, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{4}{15}\right)\) |
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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)
