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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6223, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([54,22]))
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gp:[g,chi] = znchar(Mod(3753, 6223))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6223.3753");
| Modulus: | \(6223\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(6223\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(63\) |
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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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\(\chi_{6223}(15,\cdot)\)
\(\chi_{6223}(169,\cdot)\)
\(\chi_{6223}(225,\cdot)\)
\(\chi_{6223}(288,\cdot)\)
\(\chi_{6223}(316,\cdot)\)
\(\chi_{6223}(568,\cdot)\)
\(\chi_{6223}(841,\cdot)\)
\(\chi_{6223}(1051,\cdot)\)
\(\chi_{6223}(1212,\cdot)\)
\(\chi_{6223}(1485,\cdot)\)
\(\chi_{6223}(1898,\cdot)\)
\(\chi_{6223}(1926,\cdot)\)
\(\chi_{6223}(1989,\cdot)\)
\(\chi_{6223}(2045,\cdot)\)
\(\chi_{6223}(2283,\cdot)\)
\(\chi_{6223}(2297,\cdot)\)
\(\chi_{6223}(2360,\cdot)\)
\(\chi_{6223}(2367,\cdot)\)
\(\chi_{6223}(2570,\cdot)\)
\(\chi_{6223}(3130,\cdot)\)
\(\chi_{6223}(3319,\cdot)\)
\(\chi_{6223}(3438,\cdot)\)
\(\chi_{6223}(3669,\cdot)\)
\(\chi_{6223}(3753,\cdot)\)
\(\chi_{6223}(3963,\cdot)\)
\(\chi_{6223}(4082,\cdot)\)
\(\chi_{6223}(4306,\cdot)\)
\(\chi_{6223}(4390,\cdot)\)
\(\chi_{6223}(4481,\cdot)\)
\(\chi_{6223}(4516,\cdot)\)
...
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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 };
\((5589,638)\) → \((e\left(\frac{3}{7}\right),e\left(\frac{11}{63}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 6223 }(3753, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{2}{63}\right)\) |
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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)
