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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2009, base_ring=CyclotomicField(840))
M = H._module
chi = DirichletCharacter(H, M([200,231]))
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gp:[g,chi] = znchar(Mod(151, 2009))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2009.151");
| Modulus: | \(2009\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2009\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(840\) |
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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);
|
\(\chi_{2009}(11,\cdot)\)
\(\chi_{2009}(53,\cdot)\)
\(\chi_{2009}(58,\cdot)\)
\(\chi_{2009}(60,\cdot)\)
\(\chi_{2009}(65,\cdot)\)
\(\chi_{2009}(88,\cdot)\)
\(\chi_{2009}(93,\cdot)\)
\(\chi_{2009}(95,\cdot)\)
\(\chi_{2009}(130,\cdot)\)
\(\chi_{2009}(135,\cdot)\)
\(\chi_{2009}(142,\cdot)\)
\(\chi_{2009}(149,\cdot)\)
\(\chi_{2009}(151,\cdot)\)
\(\chi_{2009}(158,\cdot)\)
\(\chi_{2009}(170,\cdot)\)
\(\chi_{2009}(179,\cdot)\)
\(\chi_{2009}(186,\cdot)\)
\(\chi_{2009}(193,\cdot)\)
\(\chi_{2009}(198,\cdot)\)
\(\chi_{2009}(212,\cdot)\)
\(\chi_{2009}(233,\cdot)\)
\(\chi_{2009}(235,\cdot)\)
\(\chi_{2009}(240,\cdot)\)
\(\chi_{2009}(261,\cdot)\)
\(\chi_{2009}(268,\cdot)\)
\(\chi_{2009}(270,\cdot)\)
\(\chi_{2009}(298,\cdot)\)
\(\chi_{2009}(317,\cdot)\)
\(\chi_{2009}(340,\cdot)\)
\(\chi_{2009}(345,\cdot)\)
...
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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 };
\((493,785)\) → \((e\left(\frac{5}{21}\right),e\left(\frac{11}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 2009 }(151, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{143}{420}\right)\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{143}{210}\right)\) | \(e\left(\frac{401}{420}\right)\) | \(e\left(\frac{197}{280}\right)\) | \(e\left(\frac{3}{140}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{293}{840}\right)\) | \(e\left(\frac{37}{840}\right)\) |
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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)
