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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(58492, base_ring=CyclotomicField(522))
M = H._module
chi = DirichletCharacter(H, M([0,174,31]))
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gp:[g,chi] = znchar(Mod(1829, 58492))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("58492.1829");
| 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: | \(14623\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(522\) |
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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 \(\chi_{14623}(1829,\cdot)\) |
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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: | 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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\(\chi_{58492}(9,\cdot)\)
\(\chi_{58492}(445,\cdot)\)
\(\chi_{58492}(557,\cdot)\)
\(\chi_{58492}(585,\cdot)\)
\(\chi_{58492}(765,\cdot)\)
\(\chi_{58492}(1129,\cdot)\)
\(\chi_{58492}(1829,\cdot)\)
\(\chi_{58492}(1901,\cdot)\)
\(\chi_{58492}(1969,\cdot)\)
\(\chi_{58492}(2629,\cdot)\)
\(\chi_{58492}(2949,\cdot)\)
\(\chi_{58492}(3021,\cdot)\)
\(\chi_{58492}(3245,\cdot)\)
\(\chi_{58492}(3553,\cdot)\)
\(\chi_{58492}(4993,\cdot)\)
\(\chi_{58492}(5261,\cdot)\)
\(\chi_{58492}(5973,\cdot)\)
\(\chi_{58492}(6409,\cdot)\)
\(\chi_{58492}(6533,\cdot)\)
\(\chi_{58492}(6633,\cdot)\)
\(\chi_{58492}(6757,\cdot)\)
\(\chi_{58492}(7009,\cdot)\)
\(\chi_{58492}(8005,\cdot)\)
\(\chi_{58492}(8089,\cdot)\)
\(\chi_{58492}(8381,\cdot)\)
\(\chi_{58492}(8969,\cdot)\)
\(\chi_{58492}(9041,\cdot)\)
\(\chi_{58492}(9209,\cdot)\)
\(\chi_{58492}(9377,\cdot)\)
\(\chi_{58492}(9517,\cdot)\)
...
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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: |
$\Q(\zeta_{261})$ |
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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 522 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((29247,50137,54321)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{31}{522}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 58492 }(1829, a) \) |
\(1\) | \(1\) | \(e\left(\frac{238}{261}\right)\) | \(e\left(\frac{79}{261}\right)\) | \(e\left(\frac{215}{261}\right)\) | \(e\left(\frac{55}{522}\right)\) | \(e\left(\frac{226}{261}\right)\) | \(e\left(\frac{56}{261}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{35}{58}\right)\) | \(e\left(\frac{158}{261}\right)\) |
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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)
