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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(104000, base_ring=CyclotomicField(1200))
M = H._module
chi = DirichletCharacter(H, M([0,975,948,200]))
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gp:[g,chi] = znchar(Mod(40213, 104000))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("104000.40213");
| Modulus: | \(104000\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(104000\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1200\) |
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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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\(\chi_{104000}(797,\cdot)\)
\(\chi_{104000}(933,\cdot)\)
\(\chi_{104000}(1037,\cdot)\)
\(\chi_{104000}(1733,\cdot)\)
\(\chi_{104000}(1837,\cdot)\)
\(\chi_{104000}(1973,\cdot)\)
\(\chi_{104000}(2077,\cdot)\)
\(\chi_{104000}(2773,\cdot)\)
\(\chi_{104000}(2877,\cdot)\)
\(\chi_{104000}(3013,\cdot)\)
\(\chi_{104000}(3117,\cdot)\)
\(\chi_{104000}(3813,\cdot)\)
\(\chi_{104000}(3917,\cdot)\)
\(\chi_{104000}(4053,\cdot)\)
\(\chi_{104000}(4853,\cdot)\)
\(\chi_{104000}(5197,\cdot)\)
\(\chi_{104000}(5997,\cdot)\)
\(\chi_{104000}(6133,\cdot)\)
\(\chi_{104000}(6237,\cdot)\)
\(\chi_{104000}(6933,\cdot)\)
\(\chi_{104000}(7037,\cdot)\)
\(\chi_{104000}(7173,\cdot)\)
\(\chi_{104000}(7277,\cdot)\)
\(\chi_{104000}(7973,\cdot)\)
\(\chi_{104000}(8077,\cdot)\)
\(\chi_{104000}(8213,\cdot)\)
\(\chi_{104000}(8317,\cdot)\)
\(\chi_{104000}(9013,\cdot)\)
\(\chi_{104000}(9117,\cdot)\)
\(\chi_{104000}(9253,\cdot)\)
...
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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: |
$\Q(\zeta_{1200})$ |
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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 1200 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((74751,58501,77377,64001)\) → \((1,e\left(\frac{13}{16}\right),e\left(\frac{79}{100}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 104000 }(40213, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{761}{1200}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{161}{600}\right)\) | \(e\left(\frac{323}{1200}\right)\) | \(e\left(\frac{113}{150}\right)\) | \(e\left(\frac{889}{1200}\right)\) | \(e\left(\frac{297}{400}\right)\) | \(e\left(\frac{319}{600}\right)\) | \(e\left(\frac{361}{400}\right)\) | \(e\left(\frac{701}{1200}\right)\) |
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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)
