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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2675, base_ring=CyclotomicField(1060))
M = H._module
chi = DirichletCharacter(H, M([159,790]))
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gp:[g,chi] = znchar(Mod(1108, 2675))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2675.1108");
| Modulus: | \(2675\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2675\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1060\) |
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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);
|
| 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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\(\chi_{2675}(2,\cdot)\)
\(\chi_{2675}(8,\cdot)\)
\(\chi_{2675}(17,\cdot)\)
\(\chi_{2675}(22,\cdot)\)
\(\chi_{2675}(28,\cdot)\)
\(\chi_{2675}(38,\cdot)\)
\(\chi_{2675}(58,\cdot)\)
\(\chi_{2675}(63,\cdot)\)
\(\chi_{2675}(67,\cdot)\)
\(\chi_{2675}(72,\cdot)\)
\(\chi_{2675}(73,\cdot)\)
\(\chi_{2675}(77,\cdot)\)
\(\chi_{2675}(78,\cdot)\)
\(\chi_{2675}(88,\cdot)\)
\(\chi_{2675}(97,\cdot)\)
\(\chi_{2675}(98,\cdot)\)
\(\chi_{2675}(103,\cdot)\)
\(\chi_{2675}(112,\cdot)\)
\(\chi_{2675}(113,\cdot)\)
\(\chi_{2675}(122,\cdot)\)
\(\chi_{2675}(127,\cdot)\)
\(\chi_{2675}(128,\cdot)\)
\(\chi_{2675}(133,\cdot)\)
\(\chi_{2675}(138,\cdot)\)
\(\chi_{2675}(152,\cdot)\)
\(\chi_{2675}(153,\cdot)\)
\(\chi_{2675}(158,\cdot)\)
\(\chi_{2675}(162,\cdot)\)
\(\chi_{2675}(167,\cdot)\)
\(\chi_{2675}(172,\cdot)\)
...
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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: |
$\Q(\zeta_{1060})$ |
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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 1060 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((1927,751)\) → \((e\left(\frac{3}{20}\right),e\left(\frac{79}{106}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 2675 }(1108, a) \) |
\(1\) | \(1\) | \(e\left(\frac{949}{1060}\right)\) | \(e\left(\frac{233}{1060}\right)\) | \(e\left(\frac{419}{530}\right)\) | \(e\left(\frac{61}{530}\right)\) | \(e\left(\frac{169}{212}\right)\) | \(e\left(\frac{727}{1060}\right)\) | \(e\left(\frac{233}{530}\right)\) | \(e\left(\frac{211}{265}\right)\) | \(e\left(\frac{11}{1060}\right)\) | \(e\left(\frac{301}{1060}\right)\) |
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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)
