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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([350,126,135]))
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gp:[g,chi] = znchar(Mod(19, 2233))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2233.19");
| 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: | 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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\(\chi_{2233}(19,\cdot)\)
\(\chi_{2233}(40,\cdot)\)
\(\chi_{2233}(61,\cdot)\)
\(\chi_{2233}(68,\cdot)\)
\(\chi_{2233}(73,\cdot)\)
\(\chi_{2233}(101,\cdot)\)
\(\chi_{2233}(171,\cdot)\)
\(\chi_{2233}(206,\cdot)\)
\(\chi_{2233}(222,\cdot)\)
\(\chi_{2233}(250,\cdot)\)
\(\chi_{2233}(271,\cdot)\)
\(\chi_{2233}(292,\cdot)\)
\(\chi_{2233}(304,\cdot)\)
\(\chi_{2233}(327,\cdot)\)
\(\chi_{2233}(369,\cdot)\)
\(\chi_{2233}(404,\cdot)\)
\(\chi_{2233}(409,\cdot)\)
\(\chi_{2233}(425,\cdot)\)
\(\chi_{2233}(437,\cdot)\)
\(\chi_{2233}(446,\cdot)\)
\(\chi_{2233}(453,\cdot)\)
\(\chi_{2233}(479,\cdot)\)
\(\chi_{2233}(514,\cdot)\)
\(\chi_{2233}(530,\cdot)\)
\(\chi_{2233}(591,\cdot)\)
\(\chi_{2233}(607,\cdot)\)
\(\chi_{2233}(612,\cdot)\)
\(\chi_{2233}(635,\cdot)\)
\(\chi_{2233}(640,\cdot)\)
\(\chi_{2233}(656,\cdot)\)
...
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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 };
\((1277,1828,2003)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{3}{10}\right),e\left(\frac{9}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 2233 }(19, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{121}{420}\right)\) | \(e\left(\frac{353}{420}\right)\) | \(e\left(\frac{121}{210}\right)\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{143}{210}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{41}{70}\right)\) |
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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)
