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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(44149, base_ring=CyclotomicField(312))
M = H._module
chi = DirichletCharacter(H, M([260,273,96]))
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gp:[g,chi] = znchar(Mod(8349, 44149))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("44149.8349");
| Modulus: | \(44149\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(6307\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(312\) |
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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: | no, induced from \(\chi_{6307}(2042,\cdot)\) |
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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: | no |
| 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);
|
\(\chi_{44149}(705,\cdot)\)
\(\chi_{44149}(950,\cdot)\)
\(\chi_{44149}(1685,\cdot)\)
\(\chi_{44149}(1844,\cdot)\)
\(\chi_{44149}(2236,\cdot)\)
\(\chi_{44149}(2824,\cdot)\)
\(\chi_{44149}(3204,\cdot)\)
\(\chi_{44149}(4282,\cdot)\)
\(\chi_{44149}(4870,\cdot)\)
\(\chi_{44149}(5262,\cdot)\)
\(\chi_{44149}(5421,\cdot)\)
\(\chi_{44149}(6989,\cdot)\)
\(\chi_{44149}(8349,\cdot)\)
\(\chi_{44149}(8508,\cdot)\)
\(\chi_{44149}(8655,\cdot)\)
\(\chi_{44149}(8900,\cdot)\)
\(\chi_{44149}(9182,\cdot)\)
\(\chi_{44149}(9341,\cdot)\)
\(\chi_{44149}(9427,\cdot)\)
\(\chi_{44149}(9586,\cdot)\)
\(\chi_{44149}(9868,\cdot)\)
\(\chi_{44149}(10848,\cdot)\)
\(\chi_{44149}(10946,\cdot)\)
\(\chi_{44149}(11007,\cdot)\)
\(\chi_{44149}(11093,\cdot)\)
\(\chi_{44149}(11154,\cdot)\)
\(\chi_{44149}(11252,\cdot)\)
\(\chi_{44149}(11779,\cdot)\)
\(\chi_{44149}(12232,\cdot)\)
\(\chi_{44149}(12820,\cdot)\)
...
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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 };
\((11714,20777,5832)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{7}{8}\right),e\left(\frac{4}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 44149 }(8349, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{35}{156}\right)\) | \(e\left(\frac{293}{312}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{1}{312}\right)\) | \(e\left(\frac{17}{104}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{137}{156}\right)\) | \(e\left(\frac{71}{312}\right)\) | \(e\left(\frac{95}{312}\right)\) | \(e\left(\frac{121}{312}\right)\) |
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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)
