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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([115,129]))
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gp:[g,chi] = znchar(Mod(338, 1251))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1251.338");
| 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: | \(138\) |
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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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\(\chi_{1251}(14,\cdot)\)
\(\chi_{1251}(23,\cdot)\)
\(\chi_{1251}(59,\cdot)\)
\(\chi_{1251}(74,\cdot)\)
\(\chi_{1251}(95,\cdot)\)
\(\chi_{1251}(149,\cdot)\)
\(\chi_{1251}(221,\cdot)\)
\(\chi_{1251}(272,\cdot)\)
\(\chi_{1251}(311,\cdot)\)
\(\chi_{1251}(317,\cdot)\)
\(\chi_{1251}(326,\cdot)\)
\(\chi_{1251}(338,\cdot)\)
\(\chi_{1251}(353,\cdot)\)
\(\chi_{1251}(362,\cdot)\)
\(\chi_{1251}(365,\cdot)\)
\(\chi_{1251}(383,\cdot)\)
\(\chi_{1251}(425,\cdot)\)
\(\chi_{1251}(479,\cdot)\)
\(\chi_{1251}(491,\cdot)\)
\(\chi_{1251}(632,\cdot)\)
\(\chi_{1251}(650,\cdot)\)
\(\chi_{1251}(659,\cdot)\)
\(\chi_{1251}(689,\cdot)\)
\(\chi_{1251}(722,\cdot)\)
\(\chi_{1251}(734,\cdot)\)
\(\chi_{1251}(743,\cdot)\)
\(\chi_{1251}(770,\cdot)\)
\(\chi_{1251}(779,\cdot)\)
\(\chi_{1251}(842,\cdot)\)
\(\chi_{1251}(848,\cdot)\)
...
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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 };
\((974,280)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{43}{46}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1251 }(338, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{69}\right)\) | \(e\left(\frac{37}{69}\right)\) | \(e\left(\frac{77}{138}\right)\) | \(e\left(\frac{5}{69}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{121}{138}\right)\) | \(e\left(\frac{34}{69}\right)\) | \(e\left(\frac{58}{69}\right)\) | \(e\left(\frac{5}{69}\right)\) |
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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)
