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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(36576, base_ring=CyclotomicField(168))
M = H._module
chi = DirichletCharacter(H, M([0,105,140,124]))
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gp:[g,chi] = znchar(Mod(16853, 36576))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("36576.16853");
| Modulus: | \(36576\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(36576\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(168\) |
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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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\(\chi_{36576}(77,\cdot)\)
\(\chi_{36576}(461,\cdot)\)
\(\chi_{36576}(1805,\cdot)\)
\(\chi_{36576}(3053,\cdot)\)
\(\chi_{36576}(4709,\cdot)\)
\(\chi_{36576}(5261,\cdot)\)
\(\chi_{36576}(5501,\cdot)\)
\(\chi_{36576}(6797,\cdot)\)
\(\chi_{36576}(7709,\cdot)\)
\(\chi_{36576}(7925,\cdot)\)
\(\chi_{36576}(8357,\cdot)\)
\(\chi_{36576}(8669,\cdot)\)
\(\chi_{36576}(9221,\cdot)\)
\(\chi_{36576}(9605,\cdot)\)
\(\chi_{36576}(10949,\cdot)\)
\(\chi_{36576}(12197,\cdot)\)
\(\chi_{36576}(13853,\cdot)\)
\(\chi_{36576}(14405,\cdot)\)
\(\chi_{36576}(14645,\cdot)\)
\(\chi_{36576}(15941,\cdot)\)
\(\chi_{36576}(16853,\cdot)\)
\(\chi_{36576}(17069,\cdot)\)
\(\chi_{36576}(17501,\cdot)\)
\(\chi_{36576}(17813,\cdot)\)
\(\chi_{36576}(18365,\cdot)\)
\(\chi_{36576}(18749,\cdot)\)
\(\chi_{36576}(20093,\cdot)\)
\(\chi_{36576}(21341,\cdot)\)
\(\chi_{36576}(22997,\cdot)\)
\(\chi_{36576}(23549,\cdot)\)
...
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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 };
\((20575,32005,8129,4321)\) → \((1,e\left(\frac{5}{8}\right),e\left(\frac{5}{6}\right),e\left(\frac{31}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 36576 }(16853, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{25}{168}\right)\) | \(e\left(\frac{71}{168}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{19}{168}\right)\) | \(e\left(\frac{13}{21}\right)\) |
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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)
