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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(32683, base_ring=CyclotomicField(462))
M = H._module
chi = DirichletCharacter(H, M([407,42,99]))
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gp:[g,chi] = znchar(Mod(1543, 32683))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("32683.1543");
| Modulus: | \(32683\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(32683\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(462\) |
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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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\(\chi_{32683}(584,\cdot)\)
\(\chi_{32683}(1223,\cdot)\)
\(\chi_{32683}(1251,\cdot)\)
\(\chi_{32683}(1405,\cdot)\)
\(\chi_{32683}(1543,\cdot)\)
\(\chi_{32683}(1720,\cdot)\)
\(\chi_{32683}(1865,\cdot)\)
\(\chi_{32683}(2005,\cdot)\)
\(\chi_{32683}(2168,\cdot)\)
\(\chi_{32683}(2362,\cdot)\)
\(\chi_{32683}(2516,\cdot)\)
\(\chi_{32683}(2672,\cdot)\)
\(\chi_{32683}(3141,\cdot)\)
\(\chi_{32683}(3937,\cdot)\)
\(\chi_{32683}(4562,\cdot)\)
\(\chi_{32683}(5204,\cdot)\)
\(\chi_{32683}(5486,\cdot)\)
\(\chi_{32683}(5983,\cdot)\)
\(\chi_{32683}(6067,\cdot)\)
\(\chi_{32683}(6268,\cdot)\)
\(\chi_{32683}(6849,\cdot)\)
\(\chi_{32683}(6935,\cdot)\)
\(\chi_{32683}(7488,\cdot)\)
\(\chi_{32683}(7852,\cdot)\)
\(\chi_{32683}(8200,\cdot)\)
\(\chi_{32683}(8270,\cdot)\)
\(\chi_{32683}(8328,\cdot)\)
\(\chi_{32683}(8825,\cdot)\)
\(\chi_{32683}(8909,\cdot)\)
\(\chi_{32683}(9110,\cdot)\)
...
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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 };
\((24013,18474,7890)\) → \((e\left(\frac{37}{42}\right),e\left(\frac{1}{11}\right),e\left(\frac{3}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 32683 }(1543, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{139}{462}\right)\) | \(e\left(\frac{94}{231}\right)\) | \(e\left(\frac{139}{231}\right)\) | \(e\left(\frac{163}{462}\right)\) | \(e\left(\frac{109}{154}\right)\) | \(e\left(\frac{139}{154}\right)\) | \(e\left(\frac{188}{231}\right)\) | \(e\left(\frac{151}{231}\right)\) | \(e\left(\frac{191}{462}\right)\) | \(e\left(\frac{2}{231}\right)\) |
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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)
