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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1599, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([60,110,63]))
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gp:[g,chi] = znchar(Mod(527, 1599))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1599.527");
| Modulus: | \(1599\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1599\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(120\) |
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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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\(\chi_{1599}(71,\cdot)\)
\(\chi_{1599}(110,\cdot)\)
\(\chi_{1599}(275,\cdot)\)
\(\chi_{1599}(293,\cdot)\)
\(\chi_{1599}(422,\cdot)\)
\(\chi_{1599}(518,\cdot)\)
\(\chi_{1599}(527,\cdot)\)
\(\chi_{1599}(557,\cdot)\)
\(\chi_{1599}(587,\cdot)\)
\(\chi_{1599}(596,\cdot)\)
\(\chi_{1599}(626,\cdot)\)
\(\chi_{1599}(704,\cdot)\)
\(\chi_{1599}(839,\cdot)\)
\(\chi_{1599}(878,\cdot)\)
\(\chi_{1599}(890,\cdot)\)
\(\chi_{1599}(908,\cdot)\)
\(\chi_{1599}(917,\cdot)\)
\(\chi_{1599}(977,\cdot)\)
\(\chi_{1599}(1055,\cdot)\)
\(\chi_{1599}(1094,\cdot)\)
\(\chi_{1599}(1142,\cdot)\)
\(\chi_{1599}(1202,\cdot)\)
\(\chi_{1599}(1241,\cdot)\)
\(\chi_{1599}(1319,\cdot)\)
\(\chi_{1599}(1406,\cdot)\)
\(\chi_{1599}(1454,\cdot)\)
\(\chi_{1599}(1493,\cdot)\)
\(\chi_{1599}(1502,\cdot)\)
\(\chi_{1599}(1532,\cdot)\)
\(\chi_{1599}(1541,\cdot)\)
...
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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 };
\((1067,1354,703)\) → \((-1,e\left(\frac{11}{12}\right),e\left(\frac{21}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 1599 }(527, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{79}{120}\right)\) |
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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)
