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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7505, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([117,52,14]))
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gp:[g,chi] = znchar(Mod(1318, 7505))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7505.1318");
| Modulus: | \(7505\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(7505\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(156\) |
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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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\(\chi_{7505}(68,\cdot)\)
\(\chi_{7505}(448,\cdot)\)
\(\chi_{7505}(638,\cdot)\)
\(\chi_{7505}(748,\cdot)\)
\(\chi_{7505}(1018,\cdot)\)
\(\chi_{7505}(1113,\cdot)\)
\(\chi_{7505}(1303,\cdot)\)
\(\chi_{7505}(1318,\cdot)\)
\(\chi_{7505}(1417,\cdot)\)
\(\chi_{7505}(1892,\cdot)\)
\(\chi_{7505}(2097,\cdot)\)
\(\chi_{7505}(2192,\cdot)\)
\(\chi_{7505}(2272,\cdot)\)
\(\chi_{7505}(2477,\cdot)\)
\(\chi_{7505}(2557,\cdot)\)
\(\chi_{7505}(2918,\cdot)\)
\(\chi_{7505}(3237,\cdot)\)
\(\chi_{7505}(3393,\cdot)\)
\(\chi_{7505}(3427,\cdot)\)
\(\chi_{7505}(3598,\cdot)\)
\(\chi_{7505}(3602,\cdot)\)
\(\chi_{7505}(3693,\cdot)\)
\(\chi_{7505}(3773,\cdot)\)
\(\chi_{7505}(3978,\cdot)\)
\(\chi_{7505}(4058,\cdot)\)
\(\chi_{7505}(4077,\cdot)\)
\(\chi_{7505}(4092,\cdot)\)
\(\chi_{7505}(4738,\cdot)\)
\(\chi_{7505}(4928,\cdot)\)
\(\chi_{7505}(5103,\cdot)\)
...
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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 };
\((1502,5531,951)\) → \((-i,e\left(\frac{1}{3}\right),e\left(\frac{7}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 7505 }(1318, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{79}{156}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{151}{156}\right)\) |
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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)
