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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(27455, base_ring=CyclotomicField(204))
M = H._module
chi = DirichletCharacter(H, M([102,117,170]))
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gp:[g,chi] = znchar(Mod(3489, 27455))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("27455.3489");
| Modulus: | \(27455\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(27455\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(204\) |
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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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\(\chi_{27455}(259,\cdot)\)
\(\chi_{27455}(939,\cdot)\)
\(\chi_{27455}(1509,\cdot)\)
\(\chi_{27455}(1874,\cdot)\)
\(\chi_{27455}(2444,\cdot)\)
\(\chi_{27455}(2554,\cdot)\)
\(\chi_{27455}(3124,\cdot)\)
\(\chi_{27455}(3489,\cdot)\)
\(\chi_{27455}(4059,\cdot)\)
\(\chi_{27455}(4169,\cdot)\)
\(\chi_{27455}(4739,\cdot)\)
\(\chi_{27455}(5104,\cdot)\)
\(\chi_{27455}(5674,\cdot)\)
\(\chi_{27455}(5784,\cdot)\)
\(\chi_{27455}(6354,\cdot)\)
\(\chi_{27455}(6719,\cdot)\)
\(\chi_{27455}(7289,\cdot)\)
\(\chi_{27455}(7399,\cdot)\)
\(\chi_{27455}(7969,\cdot)\)
\(\chi_{27455}(8334,\cdot)\)
\(\chi_{27455}(8904,\cdot)\)
\(\chi_{27455}(9014,\cdot)\)
\(\chi_{27455}(9584,\cdot)\)
\(\chi_{27455}(9949,\cdot)\)
\(\chi_{27455}(10519,\cdot)\)
\(\chi_{27455}(10629,\cdot)\)
\(\chi_{27455}(11199,\cdot)\)
\(\chi_{27455}(11564,\cdot)\)
\(\chi_{27455}(12134,\cdot)\)
\(\chi_{27455}(12244,\cdot)\)
...
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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 };
\((5492,13586,1446)\) → \((-1,e\left(\frac{39}{68}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 27455 }(3489, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{31}{102}\right)\) | \(e\left(\frac{185}{204}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{43}{204}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{83}{102}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{4}{51}\right)\) |
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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)
