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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4115, base_ring=CyclotomicField(822))
M = H._module
chi = DirichletCharacter(H, M([411,271]))
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gp:[g,chi] = znchar(Mod(24, 4115))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4115.24");
| Modulus: | \(4115\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4115\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(822\) |
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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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\(\chi_{4115}(14,\cdot)\)
\(\chi_{4115}(24,\cdot)\)
\(\chi_{4115}(44,\cdot)\)
\(\chi_{4115}(54,\cdot)\)
\(\chi_{4115}(99,\cdot)\)
\(\chi_{4115}(134,\cdot)\)
\(\chi_{4115}(149,\cdot)\)
\(\chi_{4115}(164,\cdot)\)
\(\chi_{4115}(179,\cdot)\)
\(\chi_{4115}(184,\cdot)\)
\(\chi_{4115}(229,\cdot)\)
\(\chi_{4115}(244,\cdot)\)
\(\chi_{4115}(249,\cdot)\)
\(\chi_{4115}(254,\cdot)\)
\(\chi_{4115}(259,\cdot)\)
\(\chi_{4115}(294,\cdot)\)
\(\chi_{4115}(299,\cdot)\)
\(\chi_{4115}(304,\cdot)\)
\(\chi_{4115}(309,\cdot)\)
\(\chi_{4115}(314,\cdot)\)
\(\chi_{4115}(339,\cdot)\)
\(\chi_{4115}(344,\cdot)\)
\(\chi_{4115}(354,\cdot)\)
\(\chi_{4115}(364,\cdot)\)
\(\chi_{4115}(369,\cdot)\)
\(\chi_{4115}(384,\cdot)\)
\(\chi_{4115}(399,\cdot)\)
\(\chi_{4115}(414,\cdot)\)
\(\chi_{4115}(434,\cdot)\)
\(\chi_{4115}(444,\cdot)\)
...
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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 };
\((1647,826)\) → \((-1,e\left(\frac{271}{822}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 4115 }(24, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{415}{822}\right)\) | \(e\left(\frac{341}{411}\right)\) | \(e\left(\frac{4}{411}\right)\) | \(e\left(\frac{275}{822}\right)\) | \(e\left(\frac{296}{411}\right)\) | \(e\left(\frac{141}{274}\right)\) | \(e\left(\frac{271}{411}\right)\) | \(e\left(\frac{17}{274}\right)\) | \(e\left(\frac{115}{137}\right)\) | \(e\left(\frac{107}{822}\right)\) |
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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)
