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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7913, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([76,35]))
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gp:[g,chi] = znchar(Mod(582, 7913))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7913.582");
| Modulus: | \(7913\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(7913\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(80\) |
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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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\(\chi_{7913}(166,\cdot)\)
\(\chi_{7913}(336,\cdot)\)
\(\chi_{7913}(582,\cdot)\)
\(\chi_{7913}(799,\cdot)\)
\(\chi_{7913}(938,\cdot)\)
\(\chi_{7913}(1222,\cdot)\)
\(\chi_{7913}(1673,\cdot)\)
\(\chi_{7913}(2380,\cdot)\)
\(\chi_{7913}(2506,\cdot)\)
\(\chi_{7913}(2752,\cdot)\)
\(\chi_{7913}(2831,\cdot)\)
\(\chi_{7913}(3152,\cdot)\)
\(\chi_{7913}(3278,\cdot)\)
\(\chi_{7913}(3424,\cdot)\)
\(\chi_{7913}(3524,\cdot)\)
\(\chi_{7913}(3603,\cdot)\)
\(\chi_{7913}(3670,\cdot)\)
\(\chi_{7913}(3833,\cdot)\)
\(\chi_{7913}(3887,\cdot)\)
\(\chi_{7913}(4436,\cdot)\)
\(\chi_{7913}(4682,\cdot)\)
\(\chi_{7913}(5045,\cdot)\)
\(\chi_{7913}(5817,\cdot)\)
\(\chi_{7913}(6047,\cdot)\)
\(\chi_{7913}(6319,\cdot)\)
\(\chi_{7913}(6498,\cdot)\)
\(\chi_{7913}(6565,\cdot)\)
\(\chi_{7913}(6921,\cdot)\)
\(\chi_{7913}(7091,\cdot)\)
\(\chi_{7913}(7337,\cdot)\)
...
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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 };
\((580,2707)\) → \((e\left(\frac{19}{20}\right),e\left(\frac{7}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 7913 }(582, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{40}\right)\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(1\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{73}{80}\right)\) |
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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)
