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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(20800, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([60,105,18,20]))
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gp:[g,chi] = znchar(Mod(15383, 20800))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("20800.15383");
| Modulus: | \(20800\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(10400\) |
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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 \(\chi_{10400}(8883,\cdot)\) |
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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: | 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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\(\chi_{20800}(23,\cdot)\)
\(\chi_{20800}(647,\cdot)\)
\(\chi_{20800}(823,\cdot)\)
\(\chi_{20800}(1447,\cdot)\)
\(\chi_{20800}(2103,\cdot)\)
\(\chi_{20800}(2727,\cdot)\)
\(\chi_{20800}(2903,\cdot)\)
\(\chi_{20800}(3527,\cdot)\)
\(\chi_{20800}(4183,\cdot)\)
\(\chi_{20800}(4983,\cdot)\)
\(\chi_{20800}(6263,\cdot)\)
\(\chi_{20800}(6887,\cdot)\)
\(\chi_{20800}(7063,\cdot)\)
\(\chi_{20800}(7687,\cdot)\)
\(\chi_{20800}(8967,\cdot)\)
\(\chi_{20800}(9767,\cdot)\)
\(\chi_{20800}(10423,\cdot)\)
\(\chi_{20800}(11047,\cdot)\)
\(\chi_{20800}(11223,\cdot)\)
\(\chi_{20800}(11847,\cdot)\)
\(\chi_{20800}(12503,\cdot)\)
\(\chi_{20800}(13127,\cdot)\)
\(\chi_{20800}(13303,\cdot)\)
\(\chi_{20800}(13927,\cdot)\)
\(\chi_{20800}(14583,\cdot)\)
\(\chi_{20800}(15383,\cdot)\)
\(\chi_{20800}(16663,\cdot)\)
\(\chi_{20800}(17287,\cdot)\)
\(\chi_{20800}(17463,\cdot)\)
\(\chi_{20800}(18087,\cdot)\)
...
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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()
|
\((12351,16901,14977,1601)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{3}{20}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 20800 }(15383, a) \) |
\(1\) | \(1\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{71}{120}\right)\) |
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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)
