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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4005, base_ring=CyclotomicField(264))
M = H._module
chi = DirichletCharacter(H, M([220,132,237]))
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gp:[g,chi] = znchar(Mod(1049, 4005))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4005.1049");
| Modulus: | \(4005\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4005\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(264\) |
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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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\(\chi_{4005}(14,\cdot)\)
\(\chi_{4005}(29,\cdot)\)
\(\chi_{4005}(59,\cdot)\)
\(\chi_{4005}(74,\cdot)\)
\(\chi_{4005}(104,\cdot)\)
\(\chi_{4005}(119,\cdot)\)
\(\chi_{4005}(149,\cdot)\)
\(\chi_{4005}(164,\cdot)\)
\(\chi_{4005}(209,\cdot)\)
\(\chi_{4005}(239,\cdot)\)
\(\chi_{4005}(254,\cdot)\)
\(\chi_{4005}(329,\cdot)\)
\(\chi_{4005}(389,\cdot)\)
\(\chi_{4005}(419,\cdot)\)
\(\chi_{4005}(464,\cdot)\)
\(\chi_{4005}(569,\cdot)\)
\(\chi_{4005}(599,\cdot)\)
\(\chi_{4005}(689,\cdot)\)
\(\chi_{4005}(794,\cdot)\)
\(\chi_{4005}(824,\cdot)\)
\(\chi_{4005}(839,\cdot)\)
\(\chi_{4005}(884,\cdot)\)
\(\chi_{4005}(914,\cdot)\)
\(\chi_{4005}(1049,\cdot)\)
\(\chi_{4005}(1094,\cdot)\)
\(\chi_{4005}(1109,\cdot)\)
\(\chi_{4005}(1154,\cdot)\)
\(\chi_{4005}(1184,\cdot)\)
\(\chi_{4005}(1274,\cdot)\)
\(\chi_{4005}(1289,\cdot)\)
...
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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 };
\((3116,802,181)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{79}{88}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 4005 }(1049, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{145}{264}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{215}{264}\right)\) | \(e\left(\frac{65}{264}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{37}{88}\right)\) |
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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)
