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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9025, base_ring=CyclotomicField(1710))
M = H._module
chi = DirichletCharacter(H, M([171,10]))
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gp:[g,chi] = znchar(Mod(4, 9025))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9025.4");
| 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: | \(1710\) |
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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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\(\chi_{9025}(4,\cdot)\)
\(\chi_{9025}(9,\cdot)\)
\(\chi_{9025}(44,\cdot)\)
\(\chi_{9025}(104,\cdot)\)
\(\chi_{9025}(119,\cdot)\)
\(\chi_{9025}(139,\cdot)\)
\(\chi_{9025}(169,\cdot)\)
\(\chi_{9025}(194,\cdot)\)
\(\chi_{9025}(214,\cdot)\)
\(\chi_{9025}(244,\cdot)\)
\(\chi_{9025}(264,\cdot)\)
\(\chi_{9025}(289,\cdot)\)
\(\chi_{9025}(294,\cdot)\)
\(\chi_{9025}(309,\cdot)\)
\(\chi_{9025}(329,\cdot)\)
\(\chi_{9025}(339,\cdot)\)
\(\chi_{9025}(359,\cdot)\)
\(\chi_{9025}(384,\cdot)\)
\(\chi_{9025}(404,\cdot)\)
\(\chi_{9025}(434,\cdot)\)
\(\chi_{9025}(454,\cdot)\)
\(\chi_{9025}(479,\cdot)\)
\(\chi_{9025}(484,\cdot)\)
\(\chi_{9025}(519,\cdot)\)
\(\chi_{9025}(529,\cdot)\)
\(\chi_{9025}(579,\cdot)\)
\(\chi_{9025}(594,\cdot)\)
\(\chi_{9025}(614,\cdot)\)
\(\chi_{9025}(644,\cdot)\)
\(\chi_{9025}(669,\cdot)\)
...
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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: |
$\Q(\zeta_{855})$ |
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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 1710 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((5777,3251)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{1}{171}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 9025 }(4, a) \) |
\(1\) | \(1\) | \(e\left(\frac{181}{1710}\right)\) | \(e\left(\frac{877}{1710}\right)\) | \(e\left(\frac{181}{855}\right)\) | \(e\left(\frac{529}{855}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{181}{570}\right)\) | \(e\left(\frac{22}{855}\right)\) | \(e\left(\frac{56}{285}\right)\) | \(e\left(\frac{413}{570}\right)\) | \(e\left(\frac{1409}{1710}\right)\) |
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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)
