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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4179, base_ring=CyclotomicField(198))
M = H._module
chi = DirichletCharacter(H, M([99,66,43]))
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gp:[g,chi] = znchar(Mod(590, 4179))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4179.590");
| Modulus: | \(4179\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4179\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(198\) |
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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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\(\chi_{4179}(149,\cdot)\)
\(\chi_{4179}(170,\cdot)\)
\(\chi_{4179}(179,\cdot)\)
\(\chi_{4179}(233,\cdot)\)
\(\chi_{4179}(317,\cdot)\)
\(\chi_{4179}(326,\cdot)\)
\(\chi_{4179}(389,\cdot)\)
\(\chi_{4179}(401,\cdot)\)
\(\chi_{4179}(590,\cdot)\)
\(\chi_{4179}(641,\cdot)\)
\(\chi_{4179}(674,\cdot)\)
\(\chi_{4179}(716,\cdot)\)
\(\chi_{4179}(893,\cdot)\)
\(\chi_{4179}(1010,\cdot)\)
\(\chi_{4179}(1082,\cdot)\)
\(\chi_{4179}(1103,\cdot)\)
\(\chi_{4179}(1115,\cdot)\)
\(\chi_{4179}(1124,\cdot)\)
\(\chi_{4179}(1145,\cdot)\)
\(\chi_{4179}(1178,\cdot)\)
\(\chi_{4179}(1304,\cdot)\)
\(\chi_{4179}(1346,\cdot)\)
\(\chi_{4179}(1556,\cdot)\)
\(\chi_{4179}(1691,\cdot)\)
\(\chi_{4179}(1859,\cdot)\)
\(\chi_{4179}(1934,\cdot)\)
\(\chi_{4179}(1955,\cdot)\)
\(\chi_{4179}(1964,\cdot)\)
\(\chi_{4179}(2132,\cdot)\)
\(\chi_{4179}(2195,\cdot)\)
...
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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 };
\((1394,598,799)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{43}{198}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 4179 }(590, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{198}\right)\) | \(e\left(\frac{37}{99}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{32}{99}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{35}{99}\right)\) | \(e\left(\frac{74}{99}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{11}{18}\right)\) |
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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)
