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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2592, base_ring=CyclotomicField(216))
M = H._module
chi = DirichletCharacter(H, M([108,135,88]))
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gp:[g,chi] = znchar(Mod(43, 2592))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2592.43");
| Modulus: | \(2592\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
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| Conductor: | \(2592\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(216\) |
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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: | 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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\(\chi_{2592}(43,\cdot)\)
\(\chi_{2592}(67,\cdot)\)
\(\chi_{2592}(115,\cdot)\)
\(\chi_{2592}(139,\cdot)\)
\(\chi_{2592}(187,\cdot)\)
\(\chi_{2592}(211,\cdot)\)
\(\chi_{2592}(259,\cdot)\)
\(\chi_{2592}(283,\cdot)\)
\(\chi_{2592}(331,\cdot)\)
\(\chi_{2592}(355,\cdot)\)
\(\chi_{2592}(403,\cdot)\)
\(\chi_{2592}(427,\cdot)\)
\(\chi_{2592}(475,\cdot)\)
\(\chi_{2592}(499,\cdot)\)
\(\chi_{2592}(547,\cdot)\)
\(\chi_{2592}(571,\cdot)\)
\(\chi_{2592}(619,\cdot)\)
\(\chi_{2592}(643,\cdot)\)
\(\chi_{2592}(691,\cdot)\)
\(\chi_{2592}(715,\cdot)\)
\(\chi_{2592}(763,\cdot)\)
\(\chi_{2592}(787,\cdot)\)
\(\chi_{2592}(835,\cdot)\)
\(\chi_{2592}(859,\cdot)\)
\(\chi_{2592}(907,\cdot)\)
\(\chi_{2592}(931,\cdot)\)
\(\chi_{2592}(979,\cdot)\)
\(\chi_{2592}(1003,\cdot)\)
\(\chi_{2592}(1051,\cdot)\)
\(\chi_{2592}(1075,\cdot)\)
...
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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: |
$\Q(\zeta_{216})$ |
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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 216 polynomial (not computed) |
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sage:chi.fixed_field()
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\((2431,325,1217)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{11}{27}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 2592 }(43, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{215}{216}\right)\) | \(e\left(\frac{29}{108}\right)\) | \(e\left(\frac{199}{216}\right)\) | \(e\left(\frac{137}{216}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{79}{108}\right)\) | \(e\left(\frac{107}{108}\right)\) | \(e\left(\frac{205}{216}\right)\) | \(e\left(\frac{35}{54}\right)\) |
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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)
