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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9025, base_ring=CyclotomicField(1140))
M = H._module
chi = DirichletCharacter(H, M([171,10]))
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gp:[g,chi] = znchar(Mod(8, 9025))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9025.8");
| Modulus: | \(9025\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(9025\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1140\) |
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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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\(\chi_{9025}(8,\cdot)\)
\(\chi_{9025}(12,\cdot)\)
\(\chi_{9025}(27,\cdot)\)
\(\chi_{9025}(88,\cdot)\)
\(\chi_{9025}(103,\cdot)\)
\(\chi_{9025}(122,\cdot)\)
\(\chi_{9025}(183,\cdot)\)
\(\chi_{9025}(198,\cdot)\)
\(\chi_{9025}(202,\cdot)\)
\(\chi_{9025}(217,\cdot)\)
\(\chi_{9025}(278,\cdot)\)
\(\chi_{9025}(297,\cdot)\)
\(\chi_{9025}(312,\cdot)\)
\(\chi_{9025}(373,\cdot)\)
\(\chi_{9025}(388,\cdot)\)
\(\chi_{9025}(392,\cdot)\)
\(\chi_{9025}(483,\cdot)\)
\(\chi_{9025}(487,\cdot)\)
\(\chi_{9025}(502,\cdot)\)
\(\chi_{9025}(563,\cdot)\)
\(\chi_{9025}(578,\cdot)\)
\(\chi_{9025}(597,\cdot)\)
\(\chi_{9025}(658,\cdot)\)
\(\chi_{9025}(673,\cdot)\)
\(\chi_{9025}(677,\cdot)\)
\(\chi_{9025}(692,\cdot)\)
\(\chi_{9025}(753,\cdot)\)
\(\chi_{9025}(772,\cdot)\)
\(\chi_{9025}(787,\cdot)\)
\(\chi_{9025}(848,\cdot)\)
...
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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: |
$\Q(\zeta_{1140})$ |
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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 1140 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((5777,3251)\) → \((e\left(\frac{3}{20}\right),e\left(\frac{1}{114}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 9025 }(8, a) \) |
\(1\) | \(1\) | \(e\left(\frac{181}{1140}\right)\) | \(e\left(\frac{307}{1140}\right)\) | \(e\left(\frac{181}{570}\right)\) | \(e\left(\frac{122}{285}\right)\) | \(e\left(\frac{5}{76}\right)\) | \(e\left(\frac{181}{380}\right)\) | \(e\left(\frac{307}{570}\right)\) | \(e\left(\frac{28}{95}\right)\) | \(e\left(\frac{223}{380}\right)\) | \(e\left(\frac{839}{1140}\right)\) |
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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)
