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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4663, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([100]))
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gp:[g,chi] = znchar(Mod(2215, 4663))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4663.2215");
| Modulus: | \(4663\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4663\) |
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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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\(\chi_{4663}(140,\cdot)\)
\(\chi_{4663}(288,\cdot)\)
\(\chi_{4663}(326,\cdot)\)
\(\chi_{4663}(433,\cdot)\)
\(\chi_{4663}(749,\cdot)\)
\(\chi_{4663}(870,\cdot)\)
\(\chi_{4663}(948,\cdot)\)
\(\chi_{4663}(969,\cdot)\)
\(\chi_{4663}(1194,\cdot)\)
\(\chi_{4663}(1214,\cdot)\)
\(\chi_{4663}(1238,\cdot)\)
\(\chi_{4663}(1441,\cdot)\)
\(\chi_{4663}(1446,\cdot)\)
\(\chi_{4663}(1470,\cdot)\)
\(\chi_{4663}(1494,\cdot)\)
\(\chi_{4663}(1698,\cdot)\)
\(\chi_{4663}(1892,\cdot)\)
\(\chi_{4663}(1931,\cdot)\)
\(\chi_{4663}(2215,\cdot)\)
\(\chi_{4663}(2274,\cdot)\)
\(\chi_{4663}(2323,\cdot)\)
\(\chi_{4663}(3016,\cdot)\)
\(\chi_{4663}(3024,\cdot)\)
\(\chi_{4663}(3122,\cdot)\)
\(\chi_{4663}(3143,\cdot)\)
\(\chi_{4663}(3180,\cdot)\)
\(\chi_{4663}(3406,\cdot)\)
\(\chi_{4663}(3408,\cdot)\)
\(\chi_{4663}(3421,\cdot)\)
\(\chi_{4663}(3594,\cdot)\)
...
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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 };
\(3\) → \(e\left(\frac{50}{63}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4663 }(2215, a) \) |
\(1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{63}\right)\) |
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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)
