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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([0,0,106]))
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gp:[g,chi] = znchar(Mod(8961, 58492))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("58492.8961");
| 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: | \(2089\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(261\) |
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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 \(\chi_{2089}(605,\cdot)\) |
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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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\(\chi_{58492}(29,\cdot)\)
\(\chi_{58492}(225,\cdot)\)
\(\chi_{58492}(309,\cdot)\)
\(\chi_{58492}(589,\cdot)\)
\(\chi_{58492}(841,\cdot)\)
\(\chi_{58492}(953,\cdot)\)
\(\chi_{58492}(1009,\cdot)\)
\(\chi_{58492}(1289,\cdot)\)
\(\chi_{58492}(1513,\cdot)\)
\(\chi_{58492}(1933,\cdot)\)
\(\chi_{58492}(1989,\cdot)\)
\(\chi_{58492}(2017,\cdot)\)
\(\chi_{58492}(2465,\cdot)\)
\(\chi_{58492}(2605,\cdot)\)
\(\chi_{58492}(4509,\cdot)\)
\(\chi_{58492}(5041,\cdot)\)
\(\chi_{58492}(5097,\cdot)\)
\(\chi_{58492}(5153,\cdot)\)
\(\chi_{58492}(5237,\cdot)\)
\(\chi_{58492}(5377,\cdot)\)
\(\chi_{58492}(5517,\cdot)\)
\(\chi_{58492}(6189,\cdot)\)
\(\chi_{58492}(6217,\cdot)\)
\(\chi_{58492}(6525,\cdot)\)
\(\chi_{58492}(7841,\cdot)\)
\(\chi_{58492}(7981,\cdot)\)
\(\chi_{58492}(8485,\cdot)\)
\(\chi_{58492}(8961,\cdot)\)
\(\chi_{58492}(9381,\cdot)\)
\(\chi_{58492}(9465,\cdot)\)
...
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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 261 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((29247,50137,54321)\) → \((1,1,e\left(\frac{53}{261}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 58492 }(8961, a) \) |
\(1\) | \(1\) | \(e\left(\frac{28}{261}\right)\) | \(e\left(\frac{214}{261}\right)\) | \(e\left(\frac{56}{261}\right)\) | \(e\left(\frac{167}{261}\right)\) | \(e\left(\frac{175}{261}\right)\) | \(e\left(\frac{242}{261}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{17}{87}\right)\) | \(e\left(\frac{71}{87}\right)\) | \(e\left(\frac{167}{261}\right)\) |
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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)
