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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10829, base_ring=CyclotomicField(336))
M = H._module
chi = DirichletCharacter(H, M([128,168,231]))
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gp:[g,chi] = znchar(Mod(1299, 10829))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10829.1299");
| Modulus: | \(10829\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(10829\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(336\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{10829}(142,\cdot)\)
\(\chi_{10829}(207,\cdot)\)
\(\chi_{10829}(233,\cdot)\)
\(\chi_{10829}(415,\cdot)\)
\(\chi_{10829}(571,\cdot)\)
\(\chi_{10829}(779,\cdot)\)
\(\chi_{10829}(844,\cdot)\)
\(\chi_{10829}(870,\cdot)\)
\(\chi_{10829}(1026,\cdot)\)
\(\chi_{10829}(1117,\cdot)\)
\(\chi_{10829}(1234,\cdot)\)
\(\chi_{10829}(1299,\cdot)\)
\(\chi_{10829}(1416,\cdot)\)
\(\chi_{10829}(1507,\cdot)\)
\(\chi_{10829}(1663,\cdot)\)
\(\chi_{10829}(1689,\cdot)\)
\(\chi_{10829}(1754,\cdot)\)
\(\chi_{10829}(1780,\cdot)\)
\(\chi_{10829}(1962,\cdot)\)
\(\chi_{10829}(2118,\cdot)\)
\(\chi_{10829}(2300,\cdot)\)
\(\chi_{10829}(2326,\cdot)\)
\(\chi_{10829}(2391,\cdot)\)
\(\chi_{10829}(2417,\cdot)\)
\(\chi_{10829}(2573,\cdot)\)
\(\chi_{10829}(2781,\cdot)\)
\(\chi_{10829}(2846,\cdot)\)
\(\chi_{10829}(2963,\cdot)\)
\(\chi_{10829}(3054,\cdot)\)
\(\chi_{10829}(3210,\cdot)\)
...
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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 };
\((885,834,8282)\) → \((e\left(\frac{8}{21}\right),-1,e\left(\frac{11}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 10829 }(1299, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{168}\right)\) | \(e\left(\frac{23}{336}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{331}{336}\right)\) | \(e\left(\frac{11}{112}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{23}{168}\right)\) | \(e\left(\frac{5}{336}\right)\) | \(e\left(\frac{185}{336}\right)\) | \(e\left(\frac{43}{336}\right)\) |
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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)
