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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1832, base_ring=CyclotomicField(114))
M = H._module
chi = DirichletCharacter(H, M([57,57,40]))
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gp:[g,chi] = znchar(Mod(1227, 1832))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1832.1227");
| Modulus: | \(1832\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1832\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(114\) |
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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);
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\(\chi_{1832}(3,\cdot)\)
\(\chi_{1832}(19,\cdot)\)
\(\chi_{1832}(51,\cdot)\)
\(\chi_{1832}(75,\cdot)\)
\(\chi_{1832}(83,\cdot)\)
\(\chi_{1832}(91,\cdot)\)
\(\chi_{1832}(171,\cdot)\)
\(\chi_{1832}(243,\cdot)\)
\(\chi_{1832}(355,\cdot)\)
\(\chi_{1832}(387,\cdot)\)
\(\chi_{1832}(467,\cdot)\)
\(\chi_{1832}(483,\cdot)\)
\(\chi_{1832}(539,\cdot)\)
\(\chi_{1832}(587,\cdot)\)
\(\chi_{1832}(611,\cdot)\)
\(\chi_{1832}(651,\cdot)\)
\(\chi_{1832}(675,\cdot)\)
\(\chi_{1832}(707,\cdot)\)
\(\chi_{1832}(819,\cdot)\)
\(\chi_{1832}(867,\cdot)\)
\(\chi_{1832}(883,\cdot)\)
\(\chi_{1832}(971,\cdot)\)
\(\chi_{1832}(1027,\cdot)\)
\(\chi_{1832}(1067,\cdot)\)
\(\chi_{1832}(1075,\cdot)\)
\(\chi_{1832}(1083,\cdot)\)
\(\chi_{1832}(1099,\cdot)\)
\(\chi_{1832}(1227,\cdot)\)
\(\chi_{1832}(1275,\cdot)\)
\(\chi_{1832}(1411,\cdot)\)
...
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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 };
\((1375,917,1609)\) → \((-1,-1,e\left(\frac{20}{57}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1832 }(1227, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{1}{38}\right)\) |
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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)
