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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4303, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([165,34]))
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gp:[g,chi] = znchar(Mod(1058, 4303))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4303.1058");
| Modulus: | \(4303\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4303\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(220\) |
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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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\(\chi_{4303}(18,\cdot)\)
\(\chi_{4303}(73,\cdot)\)
\(\chi_{4303}(138,\cdot)\)
\(\chi_{4303}(174,\cdot)\)
\(\chi_{4303}(187,\cdot)\)
\(\chi_{4303}(213,\cdot)\)
\(\chi_{4303}(226,\cdot)\)
\(\chi_{4303}(229,\cdot)\)
\(\chi_{4303}(343,\cdot)\)
\(\chi_{4303}(346,\cdot)\)
\(\chi_{4303}(437,\cdot)\)
\(\chi_{4303}(447,\cdot)\)
\(\chi_{4303}(473,\cdot)\)
\(\chi_{4303}(476,\cdot)\)
\(\chi_{4303}(606,\cdot)\)
\(\chi_{4303}(629,\cdot)\)
\(\chi_{4303}(840,\cdot)\)
\(\chi_{4303}(850,\cdot)\)
\(\chi_{4303}(944,\cdot)\)
\(\chi_{4303}(1045,\cdot)\)
\(\chi_{4303}(1058,\cdot)\)
\(\chi_{4303}(1071,\cdot)\)
\(\chi_{4303}(1110,\cdot)\)
\(\chi_{4303}(1152,\cdot)\)
\(\chi_{4303}(1240,\cdot)\)
\(\chi_{4303}(1256,\cdot)\)
\(\chi_{4303}(1331,\cdot)\)
\(\chi_{4303}(1334,\cdot)\)
\(\chi_{4303}(1500,\cdot)\)
\(\chi_{4303}(1529,\cdot)\)
...
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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 };
\((3642,1327)\) → \((-i,e\left(\frac{17}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4303 }(1058, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{49}{220}\right)\) | \(e\left(\frac{133}{220}\right)\) | \(e\left(\frac{169}{220}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{93}{220}\right)\) |
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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)
