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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9025, base_ring=CyclotomicField(3420))
M = H._module
chi = DirichletCharacter(H, M([1197,830]))
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gp:[g,chi] = znchar(Mod(53, 9025))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9025.53");
| 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: | \(3420\) |
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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);
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| 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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\(\chi_{9025}(2,\cdot)\)
\(\chi_{9025}(3,\cdot)\)
\(\chi_{9025}(13,\cdot)\)
\(\chi_{9025}(22,\cdot)\)
\(\chi_{9025}(33,\cdot)\)
\(\chi_{9025}(48,\cdot)\)
\(\chi_{9025}(52,\cdot)\)
\(\chi_{9025}(53,\cdot)\)
\(\chi_{9025}(67,\cdot)\)
\(\chi_{9025}(72,\cdot)\)
\(\chi_{9025}(78,\cdot)\)
\(\chi_{9025}(97,\cdot)\)
\(\chi_{9025}(98,\cdot)\)
\(\chi_{9025}(108,\cdot)\)
\(\chi_{9025}(117,\cdot)\)
\(\chi_{9025}(128,\cdot)\)
\(\chi_{9025}(147,\cdot)\)
\(\chi_{9025}(148,\cdot)\)
\(\chi_{9025}(162,\cdot)\)
\(\chi_{9025}(167,\cdot)\)
\(\chi_{9025}(173,\cdot)\)
\(\chi_{9025}(192,\cdot)\)
\(\chi_{9025}(203,\cdot)\)
\(\chi_{9025}(212,\cdot)\)
\(\chi_{9025}(222,\cdot)\)
\(\chi_{9025}(223,\cdot)\)
\(\chi_{9025}(238,\cdot)\)
\(\chi_{9025}(242,\cdot)\)
\(\chi_{9025}(287,\cdot)\)
\(\chi_{9025}(288,\cdot)\)
...
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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: |
$\Q(\zeta_{3420})$ |
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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 3420 polynomial (not computed) |
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sage:chi.fixed_field()
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\((5777,3251)\) → \((e\left(\frac{7}{20}\right),e\left(\frac{83}{342}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 9025 }(53, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2027}{3420}\right)\) | \(e\left(\frac{629}{3420}\right)\) | \(e\left(\frac{317}{1710}\right)\) | \(e\left(\frac{664}{855}\right)\) | \(e\left(\frac{35}{228}\right)\) | \(e\left(\frac{887}{1140}\right)\) | \(e\left(\frac{629}{1710}\right)\) | \(e\left(\frac{101}{285}\right)\) | \(e\left(\frac{421}{1140}\right)\) | \(e\left(\frac{1693}{3420}\right)\) |
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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)
