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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12051, base_ring=CyclotomicField(204))
M = H._module
chi = DirichletCharacter(H, M([170,153,174]))
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gp:[g,chi] = znchar(Mod(1175, 12051))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("12051.1175");
| Modulus: | \(12051\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(12051\) |
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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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\(\chi_{12051}(398,\cdot)\)
\(\chi_{12051}(434,\cdot)\)
\(\chi_{12051}(554,\cdot)\)
\(\chi_{12051}(707,\cdot)\)
\(\chi_{12051}(866,\cdot)\)
\(\chi_{12051}(1022,\cdot)\)
\(\chi_{12051}(1136,\cdot)\)
\(\chi_{12051}(1175,\cdot)\)
\(\chi_{12051}(1370,\cdot)\)
\(\chi_{12051}(1685,\cdot)\)
\(\chi_{12051}(1721,\cdot)\)
\(\chi_{12051}(1841,\cdot)\)
\(\chi_{12051}(1994,\cdot)\)
\(\chi_{12051}(2894,\cdot)\)
\(\chi_{12051}(3011,\cdot)\)
\(\chi_{12051}(3323,\cdot)\)
\(\chi_{12051}(3479,\cdot)\)
\(\chi_{12051}(3596,\cdot)\)
\(\chi_{12051}(3632,\cdot)\)
\(\chi_{12051}(3674,\cdot)\)
\(\chi_{12051}(3983,\cdot)\)
\(\chi_{12051}(4142,\cdot)\)
\(\chi_{12051}(4415,\cdot)\)
\(\chi_{12051}(4451,\cdot)\)
\(\chi_{12051}(4844,\cdot)\)
\(\chi_{12051}(4880,\cdot)\)
\(\chi_{12051}(4883,\cdot)\)
\(\chi_{12051}(5078,\cdot)\)
\(\chi_{12051}(5153,\cdot)\)
\(\chi_{12051}(5348,\cdot)\)
...
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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 };
\((5357,9271,1756)\) → \((e\left(\frac{5}{6}\right),-i,e\left(\frac{29}{34}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 12051 }(1175, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{23}{204}\right)\) | \(e\left(\frac{23}{102}\right)\) | \(e\left(\frac{157}{204}\right)\) | \(e\left(\frac{203}{204}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{23}{204}\right)\) | \(e\left(\frac{11}{102}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{12}{17}\right)\) |
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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)
