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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1691, base_ring=CyclotomicField(198))
M = H._module
chi = DirichletCharacter(H, M([143,63]))
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gp:[g,chi] = znchar(Mod(934, 1691))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1691.934");
| Modulus: | \(1691\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1691\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(198\) |
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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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\(\chi_{1691}(22,\cdot)\)
\(\chi_{1691}(146,\cdot)\)
\(\chi_{1691}(162,\cdot)\)
\(\chi_{1691}(174,\cdot)\)
\(\chi_{1691}(200,\cdot)\)
\(\chi_{1691}(203,\cdot)\)
\(\chi_{1691}(222,\cdot)\)
\(\chi_{1691}(317,\cdot)\)
\(\chi_{1691}(352,\cdot)\)
\(\chi_{1691}(413,\cdot)\)
\(\chi_{1691}(470,\cdot)\)
\(\chi_{1691}(489,\cdot)\)
\(\chi_{1691}(526,\cdot)\)
\(\chi_{1691}(545,\cdot)\)
\(\chi_{1691}(584,\cdot)\)
\(\chi_{1691}(591,\cdot)\)
\(\chi_{1691}(621,\cdot)\)
\(\chi_{1691}(648,\cdot)\)
\(\chi_{1691}(667,\cdot)\)
\(\chi_{1691}(680,\cdot)\)
\(\chi_{1691}(737,\cdot)\)
\(\chi_{1691}(756,\cdot)\)
\(\chi_{1691}(762,\cdot)\)
\(\chi_{1691}(793,\cdot)\)
\(\chi_{1691}(812,\cdot)\)
\(\chi_{1691}(851,\cdot)\)
\(\chi_{1691}(858,\cdot)\)
\(\chi_{1691}(888,\cdot)\)
\(\chi_{1691}(915,\cdot)\)
\(\chi_{1691}(934,\cdot)\)
...
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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 };
\((268,1160)\) → \((e\left(\frac{13}{18}\right),e\left(\frac{7}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1691 }(934, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{161}{198}\right)\) | \(e\left(\frac{70}{99}\right)\) | \(e\left(\frac{62}{99}\right)\) | \(e\left(\frac{82}{99}\right)\) | \(e\left(\frac{103}{198}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{41}{99}\right)\) | \(e\left(\frac{127}{198}\right)\) | \(e\left(\frac{13}{33}\right)\) |
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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)
