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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4928, base_ring=CyclotomicField(240))
M = H._module
chi = DirichletCharacter(H, M([120,45,200,48]))
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gp:[g,chi] = znchar(Mod(1027, 4928))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4928.1027");
| Modulus: | \(4928\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4928\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(240\) |
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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);
|
\(\chi_{4928}(3,\cdot)\)
\(\chi_{4928}(59,\cdot)\)
\(\chi_{4928}(75,\cdot)\)
\(\chi_{4928}(115,\cdot)\)
\(\chi_{4928}(339,\cdot)\)
\(\chi_{4928}(355,\cdot)\)
\(\chi_{4928}(411,\cdot)\)
\(\chi_{4928}(467,\cdot)\)
\(\chi_{4928}(619,\cdot)\)
\(\chi_{4928}(675,\cdot)\)
\(\chi_{4928}(691,\cdot)\)
\(\chi_{4928}(731,\cdot)\)
\(\chi_{4928}(955,\cdot)\)
\(\chi_{4928}(971,\cdot)\)
\(\chi_{4928}(1027,\cdot)\)
\(\chi_{4928}(1083,\cdot)\)
\(\chi_{4928}(1235,\cdot)\)
\(\chi_{4928}(1291,\cdot)\)
\(\chi_{4928}(1307,\cdot)\)
\(\chi_{4928}(1347,\cdot)\)
\(\chi_{4928}(1571,\cdot)\)
\(\chi_{4928}(1587,\cdot)\)
\(\chi_{4928}(1643,\cdot)\)
\(\chi_{4928}(1699,\cdot)\)
\(\chi_{4928}(1851,\cdot)\)
\(\chi_{4928}(1907,\cdot)\)
\(\chi_{4928}(1923,\cdot)\)
\(\chi_{4928}(1963,\cdot)\)
\(\chi_{4928}(2187,\cdot)\)
\(\chi_{4928}(2203,\cdot)\)
...
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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 };
\((4159,1541,2817,3137)\) → \((-1,e\left(\frac{3}{16}\right),e\left(\frac{5}{6}\right),e\left(\frac{1}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 4928 }(1027, a) \) |
\(1\) | \(1\) | \(e\left(\frac{119}{240}\right)\) | \(e\left(\frac{37}{240}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{139}{240}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{39}{80}\right)\) |
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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)
