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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3973, base_ring=CyclotomicField(476))
M = H._module
chi = DirichletCharacter(H, M([221,385]))
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gp:[g,chi] = znchar(Mod(130, 3973))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3973.130");
| Modulus: | \(3973\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3973\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(476\) |
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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);
|
\(\chi_{3973}(19,\cdot)\)
\(\chi_{3973}(39,\cdot)\)
\(\chi_{3973}(61,\cdot)\)
\(\chi_{3973}(68,\cdot)\)
\(\chi_{3973}(69,\cdot)\)
\(\chi_{3973}(76,\cdot)\)
\(\chi_{3973}(98,\cdot)\)
\(\chi_{3973}(130,\cdot)\)
\(\chi_{3973}(135,\cdot)\)
\(\chi_{3973}(156,\cdot)\)
\(\chi_{3973}(206,\cdot)\)
\(\chi_{3973}(213,\cdot)\)
\(\chi_{3973}(230,\cdot)\)
\(\chi_{3973}(235,\cdot)\)
\(\chi_{3973}(263,\cdot)\)
\(\chi_{3973}(272,\cdot)\)
\(\chi_{3973}(276,\cdot)\)
\(\chi_{3973}(282,\cdot)\)
\(\chi_{3973}(293,\cdot)\)
\(\chi_{3973}(304,\cdot)\)
\(\chi_{3973}(375,\cdot)\)
\(\chi_{3973}(379,\cdot)\)
\(\chi_{3973}(392,\cdot)\)
\(\chi_{3973}(403,\cdot)\)
\(\chi_{3973}(409,\cdot)\)
\(\chi_{3973}(443,\cdot)\)
\(\chi_{3973}(450,\cdot)\)
\(\chi_{3973}(479,\cdot)\)
\(\chi_{3973}(520,\cdot)\)
\(\chi_{3973}(540,\cdot)\)
...
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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 };
\((2467,3017)\) → \((e\left(\frac{13}{28}\right),e\left(\frac{55}{68}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 3973 }(130, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{263}{476}\right)\) | \(e\left(\frac{31}{238}\right)\) | \(e\left(\frac{25}{238}\right)\) | \(e\left(\frac{417}{476}\right)\) | \(e\left(\frac{325}{476}\right)\) | \(e\left(\frac{129}{238}\right)\) | \(e\left(\frac{313}{476}\right)\) | \(e\left(\frac{31}{119}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{135}{476}\right)\) |
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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)
