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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1251, base_ring=CyclotomicField(138))
M = H._module
chi = DirichletCharacter(H, M([46,24]))
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gp:[g,chi] = znchar(Mod(472, 1251))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1251.472");
| Modulus: | \(1251\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1251\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(69\) |
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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: | even |
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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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\(\chi_{1251}(34,\cdot)\)
\(\chi_{1251}(52,\cdot)\)
\(\chi_{1251}(79,\cdot)\)
\(\chi_{1251}(106,\cdot)\)
\(\chi_{1251}(112,\cdot)\)
\(\chi_{1251}(175,\cdot)\)
\(\chi_{1251}(184,\cdot)\)
\(\chi_{1251}(196,\cdot)\)
\(\chi_{1251}(202,\cdot)\)
\(\chi_{1251}(268,\cdot)\)
\(\chi_{1251}(322,\cdot)\)
\(\chi_{1251}(355,\cdot)\)
\(\chi_{1251}(358,\cdot)\)
\(\chi_{1251}(394,\cdot)\)
\(\chi_{1251}(403,\cdot)\)
\(\chi_{1251}(409,\cdot)\)
\(\chi_{1251}(472,\cdot)\)
\(\chi_{1251}(481,\cdot)\)
\(\chi_{1251}(508,\cdot)\)
\(\chi_{1251}(517,\cdot)\)
\(\chi_{1251}(529,\cdot)\)
\(\chi_{1251}(562,\cdot)\)
\(\chi_{1251}(592,\cdot)\)
\(\chi_{1251}(601,\cdot)\)
\(\chi_{1251}(619,\cdot)\)
\(\chi_{1251}(760,\cdot)\)
\(\chi_{1251}(772,\cdot)\)
\(\chi_{1251}(826,\cdot)\)
\(\chi_{1251}(868,\cdot)\)
\(\chi_{1251}(886,\cdot)\)
...
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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 };
\((974,280)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{4}{23}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1251 }(472, a) \) |
\(1\) | \(1\) | \(e\left(\frac{35}{69}\right)\) | \(e\left(\frac{1}{69}\right)\) | \(e\left(\frac{43}{69}\right)\) | \(e\left(\frac{2}{69}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{38}{69}\right)\) | \(e\left(\frac{55}{69}\right)\) | \(e\left(\frac{37}{69}\right)\) | \(e\left(\frac{2}{69}\right)\) |
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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)
