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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3697, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([41]))
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gp:[g,chi] = znchar(Mod(881, 3697))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3697.881");
| Modulus: | \(3697\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3697\) |
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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: | 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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\(\chi_{3697}(189,\cdot)\)
\(\chi_{3697}(200,\cdot)\)
\(\chi_{3697}(204,\cdot)\)
\(\chi_{3697}(229,\cdot)\)
\(\chi_{3697}(284,\cdot)\)
\(\chi_{3697}(365,\cdot)\)
\(\chi_{3697}(610,\cdot)\)
\(\chi_{3697}(742,\cdot)\)
\(\chi_{3697}(776,\cdot)\)
\(\chi_{3697}(802,\cdot)\)
\(\chi_{3697}(881,\cdot)\)
\(\chi_{3697}(888,\cdot)\)
\(\chi_{3697}(1154,\cdot)\)
\(\chi_{3697}(1166,\cdot)\)
\(\chi_{3697}(1189,\cdot)\)
\(\chi_{3697}(1337,\cdot)\)
\(\chi_{3697}(1523,\cdot)\)
\(\chi_{3697}(1539,\cdot)\)
\(\chi_{3697}(1596,\cdot)\)
\(\chi_{3697}(1792,\cdot)\)
\(\chi_{3697}(1905,\cdot)\)
\(\chi_{3697}(2101,\cdot)\)
\(\chi_{3697}(2158,\cdot)\)
\(\chi_{3697}(2174,\cdot)\)
\(\chi_{3697}(2360,\cdot)\)
\(\chi_{3697}(2508,\cdot)\)
\(\chi_{3697}(2531,\cdot)\)
\(\chi_{3697}(2543,\cdot)\)
\(\chi_{3697}(2809,\cdot)\)
\(\chi_{3697}(2816,\cdot)\)
...
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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 };
\(5\) → \(e\left(\frac{41}{132}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 3697 }(881, a) \) |
\(1\) | \(1\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{14}{33}\right)\) |
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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)
