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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1045, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([135,54,20]))
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gp:[g,chi] = znchar(Mod(118, 1045))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1045.118");
| Modulus: | \(1045\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1045\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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);
|
\(\chi_{1045}(17,\cdot)\)
\(\chi_{1045}(28,\cdot)\)
\(\chi_{1045}(62,\cdot)\)
\(\chi_{1045}(63,\cdot)\)
\(\chi_{1045}(73,\cdot)\)
\(\chi_{1045}(112,\cdot)\)
\(\chi_{1045}(118,\cdot)\)
\(\chi_{1045}(123,\cdot)\)
\(\chi_{1045}(138,\cdot)\)
\(\chi_{1045}(233,\cdot)\)
\(\chi_{1045}(237,\cdot)\)
\(\chi_{1045}(272,\cdot)\)
\(\chi_{1045}(282,\cdot)\)
\(\chi_{1045}(283,\cdot)\)
\(\chi_{1045}(327,\cdot)\)
\(\chi_{1045}(332,\cdot)\)
\(\chi_{1045}(347,\cdot)\)
\(\chi_{1045}(348,\cdot)\)
\(\chi_{1045}(358,\cdot)\)
\(\chi_{1045}(403,\cdot)\)
\(\chi_{1045}(442,\cdot)\)
\(\chi_{1045}(453,\cdot)\)
\(\chi_{1045}(492,\cdot)\)
\(\chi_{1045}(503,\cdot)\)
\(\chi_{1045}(557,\cdot)\)
\(\chi_{1045}(567,\cdot)\)
\(\chi_{1045}(568,\cdot)\)
\(\chi_{1045}(612,\cdot)\)
\(\chi_{1045}(613,\cdot)\)
\(\chi_{1045}(633,\cdot)\)
...
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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 };
\((837,761,496)\) → \((-i,e\left(\frac{3}{10}\right),e\left(\frac{1}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 1045 }(118, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{180}\right)\) | \(e\left(\frac{17}{180}\right)\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{23}{90}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{17}{90}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{19}{180}\right)\) | \(e\left(\frac{61}{90}\right)\) |
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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)
