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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(65065, base_ring=CyclotomicField(780))
M = H._module
chi = DirichletCharacter(H, M([195,520,546,380]))
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gp:[g,chi] = znchar(Mod(10182, 65065))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("65065.10182");
| Modulus: | \(65065\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(65065\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(780\) |
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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);
|
\(\chi_{65065}(172,\cdot)\)
\(\chi_{65065}(282,\cdot)\)
\(\chi_{65065}(1173,\cdot)\)
\(\chi_{65065}(1283,\cdot)\)
\(\chi_{65065}(1537,\cdot)\)
\(\chi_{65065}(1647,\cdot)\)
\(\chi_{65065}(2538,\cdot)\)
\(\chi_{65065}(2648,\cdot)\)
\(\chi_{65065}(3467,\cdot)\)
\(\chi_{65065}(3812,\cdot)\)
\(\chi_{65065}(3922,\cdot)\)
\(\chi_{65065}(4358,\cdot)\)
\(\chi_{65065}(4468,\cdot)\)
\(\chi_{65065}(4813,\cdot)\)
\(\chi_{65065}(5177,\cdot)\)
\(\chi_{65065}(5287,\cdot)\)
\(\chi_{65065}(6178,\cdot)\)
\(\chi_{65065}(6288,\cdot)\)
\(\chi_{65065}(6542,\cdot)\)
\(\chi_{65065}(6652,\cdot)\)
\(\chi_{65065}(7543,\cdot)\)
\(\chi_{65065}(7653,\cdot)\)
\(\chi_{65065}(8362,\cdot)\)
\(\chi_{65065}(8817,\cdot)\)
\(\chi_{65065}(8927,\cdot)\)
\(\chi_{65065}(9363,\cdot)\)
\(\chi_{65065}(9473,\cdot)\)
\(\chi_{65065}(9818,\cdot)\)
\(\chi_{65065}(9928,\cdot)\)
\(\chi_{65065}(10182,\cdot)\)
...
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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 };
\((26027,46476,41406,6931)\) → \((i,e\left(\frac{2}{3}\right),e\left(\frac{7}{10}\right),e\left(\frac{19}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(16\) | \(17\) | \(18\) |
| \( \chi_{ 65065 }(10182, a) \) |
\(1\) | \(1\) | \(e\left(\frac{601}{780}\right)\) | \(e\left(\frac{111}{260}\right)\) | \(e\left(\frac{211}{390}\right)\) | \(e\left(\frac{77}{390}\right)\) | \(e\left(\frac{81}{260}\right)\) | \(e\left(\frac{111}{130}\right)\) | \(e\left(\frac{151}{156}\right)\) | \(e\left(\frac{16}{195}\right)\) | \(e\left(\frac{269}{780}\right)\) | \(e\left(\frac{487}{780}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
