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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2041, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([13,4]))
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gp:[g,chi] = znchar(Mod(782, 2041))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2041.782");
| Modulus: | \(2041\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2041\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(156\) |
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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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\(\chi_{2041}(11,\cdot)\)
\(\chi_{2041}(71,\cdot)\)
\(\chi_{2041}(89,\cdot)\)
\(\chi_{2041}(115,\cdot)\)
\(\chi_{2041}(124,\cdot)\)
\(\chi_{2041}(154,\cdot)\)
\(\chi_{2041}(197,\cdot)\)
\(\chi_{2041}(228,\cdot)\)
\(\chi_{2041}(344,\cdot)\)
\(\chi_{2041}(349,\cdot)\)
\(\chi_{2041}(427,\cdot)\)
\(\chi_{2041}(435,\cdot)\)
\(\chi_{2041}(501,\cdot)\)
\(\chi_{2041}(592,\cdot)\)
\(\chi_{2041}(639,\cdot)\)
\(\chi_{2041}(717,\cdot)\)
\(\chi_{2041}(734,\cdot)\)
\(\chi_{2041}(743,\cdot)\)
\(\chi_{2041}(760,\cdot)\)
\(\chi_{2041}(782,\cdot)\)
\(\chi_{2041}(804,\cdot)\)
\(\chi_{2041}(891,\cdot)\)
\(\chi_{2041}(917,\cdot)\)
\(\chi_{2041}(977,\cdot)\)
\(\chi_{2041}(994,\cdot)\)
\(\chi_{2041}(1051,\cdot)\)
\(\chi_{2041}(1055,\cdot)\)
\(\chi_{2041}(1116,\cdot)\)
\(\chi_{2041}(1151,\cdot)\)
\(\chi_{2041}(1246,\cdot)\)
...
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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 };
\((158,1418)\) → \((e\left(\frac{1}{12}\right),e\left(\frac{1}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 2041 }(782, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{109}{156}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{121}{156}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{107}{156}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{47}{156}\right)\) |
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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)
