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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(26912, base_ring=CyclotomicField(812))
M = H._module
chi = DirichletCharacter(H, M([406,203,248]))
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gp:[g,chi] = znchar(Mod(1383, 26912))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("26912.1383");
| Modulus: | \(26912\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(13456\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(812\) |
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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: | no, induced from \(\chi_{13456}(4747,\cdot)\) |
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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: | no |
| 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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\(\chi_{26912}(7,\cdot)\)
\(\chi_{26912}(23,\cdot)\)
\(\chi_{26912}(103,\cdot)\)
\(\chi_{26912}(199,\cdot)\)
\(\chi_{26912}(343,\cdot)\)
\(\chi_{26912}(455,\cdot)\)
\(\chi_{26912}(471,\cdot)\)
\(\chi_{26912}(487,\cdot)\)
\(\chi_{26912}(567,\cdot)\)
\(\chi_{26912}(663,\cdot)\)
\(\chi_{26912}(807,\cdot)\)
\(\chi_{26912}(919,\cdot)\)
\(\chi_{26912}(935,\cdot)\)
\(\chi_{26912}(951,\cdot)\)
\(\chi_{26912}(1127,\cdot)\)
\(\chi_{26912}(1271,\cdot)\)
\(\chi_{26912}(1383,\cdot)\)
\(\chi_{26912}(1399,\cdot)\)
\(\chi_{26912}(1495,\cdot)\)
\(\chi_{26912}(1591,\cdot)\)
\(\chi_{26912}(1735,\cdot)\)
\(\chi_{26912}(1847,\cdot)\)
\(\chi_{26912}(1863,\cdot)\)
\(\chi_{26912}(1879,\cdot)\)
\(\chi_{26912}(1959,\cdot)\)
\(\chi_{26912}(2055,\cdot)\)
\(\chi_{26912}(2199,\cdot)\)
\(\chi_{26912}(2311,\cdot)\)
\(\chi_{26912}(2343,\cdot)\)
\(\chi_{26912}(2423,\cdot)\)
...
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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 };
\((11775,3365,5889)\) → \((-1,i,e\left(\frac{62}{203}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 26912 }(1383, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{211}{812}\right)\) | \(e\left(\frac{395}{812}\right)\) | \(e\left(\frac{170}{203}\right)\) | \(e\left(\frac{211}{406}\right)\) | \(e\left(\frac{397}{812}\right)\) | \(e\left(\frac{761}{812}\right)\) | \(e\left(\frac{303}{406}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{111}{812}\right)\) | \(e\left(\frac{79}{812}\right)\) |
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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)
