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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3335, base_ring=CyclotomicField(154))
M = H._module
chi = DirichletCharacter(H, M([77,98,143]))
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gp:[g,chi] = znchar(Mod(979, 3335))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3335.979");
| Modulus: | \(3335\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3335\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(154\) |
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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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\(\chi_{3335}(4,\cdot)\)
\(\chi_{3335}(9,\cdot)\)
\(\chi_{3335}(64,\cdot)\)
\(\chi_{3335}(154,\cdot)\)
\(\chi_{3335}(179,\cdot)\)
\(\chi_{3335}(209,\cdot)\)
\(\chi_{3335}(294,\cdot)\)
\(\chi_{3335}(324,\cdot)\)
\(\chi_{3335}(354,\cdot)\)
\(\chi_{3335}(399,\cdot)\)
\(\chi_{3335}(439,\cdot)\)
\(\chi_{3335}(469,\cdot)\)
\(\chi_{3335}(499,\cdot)\)
\(\chi_{3335}(564,\cdot)\)
\(\chi_{3335}(584,\cdot)\)
\(\chi_{3335}(614,\cdot)\)
\(\chi_{3335}(729,\cdot)\)
\(\chi_{3335}(834,\cdot)\)
\(\chi_{3335}(854,\cdot)\)
\(\chi_{3335}(979,\cdot)\)
\(\chi_{3335}(1024,\cdot)\)
\(\chi_{3335}(1269,\cdot)\)
\(\chi_{3335}(1314,\cdot)\)
\(\chi_{3335}(1369,\cdot)\)
\(\chi_{3335}(1434,\cdot)\)
\(\chi_{3335}(1484,\cdot)\)
\(\chi_{3335}(1559,\cdot)\)
\(\chi_{3335}(1599,\cdot)\)
\(\chi_{3335}(1659,\cdot)\)
\(\chi_{3335}(1704,\cdot)\)
...
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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 };
\((2002,2466,3221)\) → \((-1,e\left(\frac{7}{11}\right),e\left(\frac{13}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 3335 }(979, a) \) |
\(1\) | \(1\) | \(e\left(\frac{54}{77}\right)\) | \(e\left(\frac{25}{77}\right)\) | \(e\left(\frac{31}{77}\right)\) | \(e\left(\frac{2}{77}\right)\) | \(e\left(\frac{113}{154}\right)\) | \(e\left(\frac{8}{77}\right)\) | \(e\left(\frac{50}{77}\right)\) | \(e\left(\frac{145}{154}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{19}{154}\right)\) |
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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)
