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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(58492, base_ring=CyclotomicField(2088))
M = H._module
chi = DirichletCharacter(H, M([1044,1740,691]))
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gp:[g,chi] = znchar(Mod(607, 58492))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("58492.607");
| Modulus: | \(58492\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(58492\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(2088\) |
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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);
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| 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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\(\chi_{58492}(31,\cdot)\)
\(\chi_{58492}(59,\cdot)\)
\(\chi_{58492}(115,\cdot)\)
\(\chi_{58492}(131,\cdot)\)
\(\chi_{58492}(227,\cdot)\)
\(\chi_{58492}(271,\cdot)\)
\(\chi_{58492}(283,\cdot)\)
\(\chi_{58492}(439,\cdot)\)
\(\chi_{58492}(495,\cdot)\)
\(\chi_{58492}(591,\cdot)\)
\(\chi_{58492}(607,\cdot)\)
\(\chi_{58492}(635,\cdot)\)
\(\chi_{58492}(747,\cdot)\)
\(\chi_{58492}(831,\cdot)\)
\(\chi_{58492}(871,\cdot)\)
\(\chi_{58492}(887,\cdot)\)
\(\chi_{58492}(1139,\cdot)\)
\(\chi_{58492}(1179,\cdot)\)
\(\chi_{58492}(1223,\cdot)\)
\(\chi_{58492}(1307,\cdot)\)
\(\chi_{58492}(1543,\cdot)\)
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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: |
$\Q(\zeta_{2088})$ |
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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 2088 polynomial (not computed) |
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sage:chi.fixed_field()
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\((29247,50137,54321)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{691}{2088}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 58492 }(607, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{565}{1044}\right)\) | \(e\left(\frac{397}{1044}\right)\) | \(e\left(\frac{43}{522}\right)\) | \(e\left(\frac{805}{2088}\right)\) | \(e\left(\frac{247}{1044}\right)\) | \(e\left(\frac{481}{522}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{25}{232}\right)\) | \(e\left(\frac{159}{232}\right)\) | \(e\left(\frac{397}{522}\right)\) |
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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)
