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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4663, base_ring=CyclotomicField(666))
M = H._module
chi = DirichletCharacter(H, M([604]))
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gp:[g,chi] = znchar(Mod(101, 4663))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4663.101");
| 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: | \(333\) |
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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);
|
\(\chi_{4663}(53,\cdot)\)
\(\chi_{4663}(58,\cdot)\)
\(\chi_{4663}(76,\cdot)\)
\(\chi_{4663}(88,\cdot)\)
\(\chi_{4663}(98,\cdot)\)
\(\chi_{4663}(101,\cdot)\)
\(\chi_{4663}(119,\cdot)\)
\(\chi_{4663}(124,\cdot)\)
\(\chi_{4663}(163,\cdot)\)
\(\chi_{4663}(172,\cdot)\)
\(\chi_{4663}(197,\cdot)\)
\(\chi_{4663}(205,\cdot)\)
\(\chi_{4663}(240,\cdot)\)
\(\chi_{4663}(244,\cdot)\)
\(\chi_{4663}(278,\cdot)\)
\(\chi_{4663}(342,\cdot)\)
\(\chi_{4663}(355,\cdot)\)
\(\chi_{4663}(368,\cdot)\)
\(\chi_{4663}(377,\cdot)\)
\(\chi_{4663}(389,\cdot)\)
\(\chi_{4663}(396,\cdot)\)
\(\chi_{4663}(402,\cdot)\)
\(\chi_{4663}(450,\cdot)\)
\(\chi_{4663}(461,\cdot)\)
\(\chi_{4663}(474,\cdot)\)
\(\chi_{4663}(494,\cdot)\)
\(\chi_{4663}(501,\cdot)\)
\(\chi_{4663}(531,\cdot)\)
\(\chi_{4663}(558,\cdot)\)
\(\chi_{4663}(572,\cdot)\)
...
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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 };
\(3\) → \(e\left(\frac{302}{333}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4663 }(101, a) \) |
\(1\) | \(1\) | \(e\left(\frac{88}{111}\right)\) | \(e\left(\frac{302}{333}\right)\) | \(e\left(\frac{65}{111}\right)\) | \(e\left(\frac{36}{37}\right)\) | \(e\left(\frac{233}{333}\right)\) | \(e\left(\frac{295}{333}\right)\) | \(e\left(\frac{14}{37}\right)\) | \(e\left(\frac{271}{333}\right)\) | \(e\left(\frac{85}{111}\right)\) | \(e\left(\frac{325}{333}\right)\) |
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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)
