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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2233, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([350,84,255]))
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gp:[g,chi] = znchar(Mod(1181, 2233))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2233.1181");
| Modulus: | \(2233\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2233\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(420\) |
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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);
|
\(\chi_{2233}(3,\cdot)\)
\(\chi_{2233}(26,\cdot)\)
\(\chi_{2233}(31,\cdot)\)
\(\chi_{2233}(47,\cdot)\)
\(\chi_{2233}(108,\cdot)\)
\(\chi_{2233}(124,\cdot)\)
\(\chi_{2233}(159,\cdot)\)
\(\chi_{2233}(185,\cdot)\)
\(\chi_{2233}(192,\cdot)\)
\(\chi_{2233}(201,\cdot)\)
\(\chi_{2233}(213,\cdot)\)
\(\chi_{2233}(229,\cdot)\)
\(\chi_{2233}(234,\cdot)\)
\(\chi_{2233}(269,\cdot)\)
\(\chi_{2233}(311,\cdot)\)
\(\chi_{2233}(334,\cdot)\)
\(\chi_{2233}(346,\cdot)\)
\(\chi_{2233}(367,\cdot)\)
\(\chi_{2233}(388,\cdot)\)
\(\chi_{2233}(416,\cdot)\)
\(\chi_{2233}(432,\cdot)\)
\(\chi_{2233}(467,\cdot)\)
\(\chi_{2233}(537,\cdot)\)
\(\chi_{2233}(565,\cdot)\)
\(\chi_{2233}(570,\cdot)\)
\(\chi_{2233}(577,\cdot)\)
\(\chi_{2233}(598,\cdot)\)
\(\chi_{2233}(619,\cdot)\)
\(\chi_{2233}(675,\cdot)\)
\(\chi_{2233}(698,\cdot)\)
...
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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 };
\((1277,1828,2003)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{5}\right),e\left(\frac{17}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 2233 }(1181, a) \) |
\(1\) | \(1\) | \(e\left(\frac{199}{420}\right)\) | \(e\left(\frac{197}{420}\right)\) | \(e\left(\frac{199}{210}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{59}{140}\right)\) | \(e\left(\frac{197}{210}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{22}{35}\right)\) |
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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)
