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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([1007,960]))
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gp:[g,chi] = znchar(Mod(1063, 2675))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2675.1063");
| 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: | 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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\(\chi_{2675}(3,\cdot)\)
\(\chi_{2675}(12,\cdot)\)
\(\chi_{2675}(13,\cdot)\)
\(\chi_{2675}(23,\cdot)\)
\(\chi_{2675}(27,\cdot)\)
\(\chi_{2675}(33,\cdot)\)
\(\chi_{2675}(37,\cdot)\)
\(\chi_{2675}(42,\cdot)\)
\(\chi_{2675}(47,\cdot)\)
\(\chi_{2675}(48,\cdot)\)
\(\chi_{2675}(52,\cdot)\)
\(\chi_{2675}(53,\cdot)\)
\(\chi_{2675}(62,\cdot)\)
\(\chi_{2675}(83,\cdot)\)
\(\chi_{2675}(87,\cdot)\)
\(\chi_{2675}(92,\cdot)\)
\(\chi_{2675}(102,\cdot)\)
\(\chi_{2675}(117,\cdot)\)
\(\chi_{2675}(123,\cdot)\)
\(\chi_{2675}(137,\cdot)\)
\(\chi_{2675}(142,\cdot)\)
\(\chi_{2675}(147,\cdot)\)
\(\chi_{2675}(148,\cdot)\)
\(\chi_{2675}(163,\cdot)\)
\(\chi_{2675}(183,\cdot)\)
\(\chi_{2675}(188,\cdot)\)
\(\chi_{2675}(192,\cdot)\)
\(\chi_{2675}(197,\cdot)\)
\(\chi_{2675}(208,\cdot)\)
\(\chi_{2675}(212,\cdot)\)
...
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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()
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\((1927,751)\) → \((e\left(\frac{19}{20}\right),e\left(\frac{48}{53}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 2675 }(1063, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{907}{1060}\right)\) | \(e\left(\frac{49}{1060}\right)\) | \(e\left(\frac{377}{530}\right)\) | \(e\left(\frac{239}{265}\right)\) | \(e\left(\frac{147}{212}\right)\) | \(e\left(\frac{601}{1060}\right)\) | \(e\left(\frac{49}{530}\right)\) | \(e\left(\frac{33}{265}\right)\) | \(e\left(\frac{803}{1060}\right)\) | \(e\left(\frac{773}{1060}\right)\) |
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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)
