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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(19475, base_ring=CyclotomicField(360))
M = H._module
chi = DirichletCharacter(H, M([342,160,9]))
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gp:[g,chi] = znchar(Mod(3163, 19475))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("19475.3163");
| Modulus: | \(19475\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(19475\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(360\) |
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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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\(\chi_{19475}(473,\cdot)\)
\(\chi_{19475}(522,\cdot)\)
\(\chi_{19475}(587,\cdot)\)
\(\chi_{19475}(803,\cdot)\)
\(\chi_{19475}(872,\cdot)\)
\(\chi_{19475}(878,\cdot)\)
\(\chi_{19475}(1012,\cdot)\)
\(\chi_{19475}(1498,\cdot)\)
\(\chi_{19475}(1612,\cdot)\)
\(\chi_{19475}(1792,\cdot)\)
\(\chi_{19475}(1828,\cdot)\)
\(\chi_{19475}(1852,\cdot)\)
\(\chi_{19475}(1897,\cdot)\)
\(\chi_{19475}(2037,\cdot)\)
\(\chi_{19475}(2267,\cdot)\)
\(\chi_{19475}(2817,\cdot)\)
\(\chi_{19475}(2848,\cdot)\)
\(\chi_{19475}(3008,\cdot)\)
\(\chi_{19475}(3027,\cdot)\)
\(\chi_{19475}(3163,\cdot)\)
\(\chi_{19475}(3292,\cdot)\)
\(\chi_{19475}(3597,\cdot)\)
\(\chi_{19475}(3842,\cdot)\)
\(\chi_{19475}(3873,\cdot)\)
\(\chi_{19475}(4033,\cdot)\)
\(\chi_{19475}(4052,\cdot)\)
\(\chi_{19475}(4208,\cdot)\)
\(\chi_{19475}(4317,\cdot)\)
\(\chi_{19475}(4463,\cdot)\)
\(\chi_{19475}(4622,\cdot)\)
...
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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 };
\((14802,18451,9026)\) → \((e\left(\frac{19}{20}\right),e\left(\frac{4}{9}\right),e\left(\frac{1}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 19475 }(3163, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{289}{360}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{109}{180}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{17}{360}\right)\) |
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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)
