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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4355, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([99,66,76]))
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gp:[g,chi] = znchar(Mod(2313, 4355))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4355.2313");
| Modulus: | \(4355\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4355\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(132\) |
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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: | odd |
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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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\(\chi_{4355}(77,\cdot)\)
\(\chi_{4355}(103,\cdot)\)
\(\chi_{4355}(207,\cdot)\)
\(\chi_{4355}(272,\cdot)\)
\(\chi_{4355}(428,\cdot)\)
\(\chi_{4355}(467,\cdot)\)
\(\chi_{4355}(753,\cdot)\)
\(\chi_{4355}(792,\cdot)\)
\(\chi_{4355}(948,\cdot)\)
\(\chi_{4355}(987,\cdot)\)
\(\chi_{4355}(1052,\cdot)\)
\(\chi_{4355}(1078,\cdot)\)
\(\chi_{4355}(1143,\cdot)\)
\(\chi_{4355}(1312,\cdot)\)
\(\chi_{4355}(1338,\cdot)\)
\(\chi_{4355}(1442,\cdot)\)
\(\chi_{4355}(1507,\cdot)\)
\(\chi_{4355}(1663,\cdot)\)
\(\chi_{4355}(1832,\cdot)\)
\(\chi_{4355}(1858,\cdot)\)
\(\chi_{4355}(1897,\cdot)\)
\(\chi_{4355}(1923,\cdot)\)
\(\chi_{4355}(1962,\cdot)\)
\(\chi_{4355}(2027,\cdot)\)
\(\chi_{4355}(2183,\cdot)\)
\(\chi_{4355}(2313,\cdot)\)
\(\chi_{4355}(2378,\cdot)\)
\(\chi_{4355}(2703,\cdot)\)
\(\chi_{4355}(2768,\cdot)\)
\(\chi_{4355}(2807,\cdot)\)
...
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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 };
\((872,1341,2146)\) → \((-i,-1,e\left(\frac{19}{33}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 4355 }(2313, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{109}{132}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{65}{132}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{47}{132}\right)\) | \(e\left(\frac{7}{22}\right)\) |
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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)
