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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8027, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([102,77]))
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gp:[g,chi] = znchar(Mod(1372, 8027))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8027.1372");
| Modulus: | \(8027\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8027\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(132\) |
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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);
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\(\chi_{8027}(189,\cdot)\)
\(\chi_{8027}(373,\cdot)\)
\(\chi_{8027}(674,\cdot)\)
\(\chi_{8027}(858,\cdot)\)
\(\chi_{8027}(1023,\cdot)\)
\(\chi_{8027}(1207,\cdot)\)
\(\chi_{8027}(1236,\cdot)\)
\(\chi_{8027}(1372,\cdot)\)
\(\chi_{8027}(1420,\cdot)\)
\(\chi_{8027}(1556,\cdot)\)
\(\chi_{8027}(1585,\cdot)\)
\(\chi_{8027}(1721,\cdot)\)
\(\chi_{8027}(1769,\cdot)\)
\(\chi_{8027}(1905,\cdot)\)
\(\chi_{8027}(2632,\cdot)\)
\(\chi_{8027}(2816,\cdot)\)
\(\chi_{8027}(2981,\cdot)\)
\(\chi_{8027}(3165,\cdot)\)
\(\chi_{8027}(3815,\cdot)\)
\(\chi_{8027}(3999,\cdot)\)
\(\chi_{8027}(4377,\cdot)\)
\(\chi_{8027}(4513,\cdot)\)
\(\chi_{8027}(4561,\cdot)\)
\(\chi_{8027}(4697,\cdot)\)
\(\chi_{8027}(4726,\cdot)\)
\(\chi_{8027}(4910,\cdot)\)
\(\chi_{8027}(5075,\cdot)\)
\(\chi_{8027}(5259,\cdot)\)
\(\chi_{8027}(5424,\cdot)\)
\(\chi_{8027}(5560,\cdot)\)
...
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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 };
\((350,5935)\) → \((e\left(\frac{17}{22}\right),e\left(\frac{7}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8027 }(1372, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{101}{132}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) |
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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)
