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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6293, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([105,60,203]))
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gp:[g,chi] = znchar(Mod(517, 6293))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6293.517");
| Modulus: | \(6293\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(6293\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(210\) |
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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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\(\chi_{6293}(83,\cdot)\)
\(\chi_{6293}(384,\cdot)\)
\(\chi_{6293}(489,\cdot)\)
\(\chi_{6293}(517,\cdot)\)
\(\chi_{6293}(538,\cdot)\)
\(\chi_{6293}(923,\cdot)\)
\(\chi_{6293}(951,\cdot)\)
\(\chi_{6293}(1035,\cdot)\)
\(\chi_{6293}(1098,\cdot)\)
\(\chi_{6293}(1350,\cdot)\)
\(\chi_{6293}(1357,\cdot)\)
\(\chi_{6293}(1532,\cdot)\)
\(\chi_{6293}(1553,\cdot)\)
\(\chi_{6293}(1602,\cdot)\)
\(\chi_{6293}(1966,\cdot)\)
\(\chi_{6293}(2008,\cdot)\)
\(\chi_{6293}(2253,\cdot)\)
\(\chi_{6293}(2316,\cdot)\)
\(\chi_{6293}(2617,\cdot)\)
\(\chi_{6293}(2659,\cdot)\)
\(\chi_{6293}(2750,\cdot)\)
\(\chi_{6293}(2771,\cdot)\)
\(\chi_{6293}(2925,\cdot)\)
\(\chi_{6293}(3184,\cdot)\)
\(\chi_{6293}(3268,\cdot)\)
\(\chi_{6293}(3359,\cdot)\)
\(\chi_{6293}(3737,\cdot)\)
\(\chi_{6293}(3793,\cdot)\)
\(\chi_{6293}(3835,\cdot)\)
\(\chi_{6293}(3940,\cdot)\)
...
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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 };
\((2698,5860,4467)\) → \((-1,e\left(\frac{2}{7}\right),e\left(\frac{29}{30}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 6293 }(517, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{127}{210}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{13}{15}\right)\) |
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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)
