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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(256025, base_ring=CyclotomicField(1260))
M = H._module
chi = DirichletCharacter(H, M([819,660,756,980]))
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gp:[g,chi] = znchar(Mod(19842, 256025))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("256025.19842");
| Modulus: | \(256025\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(256025\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1260\) |
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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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\(\chi_{256025}(522,\cdot)\)
\(\chi_{256025}(1213,\cdot)\)
\(\chi_{256025}(1423,\cdot)\)
\(\chi_{256025}(1852,\cdot)\)
\(\chi_{256025}(5058,\cdot)\)
\(\chi_{256025}(7198,\cdot)\)
\(\chi_{256025}(7473,\cdot)\)
\(\chi_{256025}(7947,\cdot)\)
\(\chi_{256025}(9277,\cdot)\)
\(\chi_{256025}(9942,\cdot)\)
\(\chi_{256025}(10378,\cdot)\)
\(\chi_{256025}(11797,\cdot)\)
\(\chi_{256025}(13038,\cdot)\)
\(\chi_{256025}(13127,\cdot)\)
\(\chi_{256025}(13792,\cdot)\)
\(\chi_{256025}(17812,\cdot)\)
\(\chi_{256025}(17847,\cdot)\)
\(\chi_{256025}(18078,\cdot)\)
\(\chi_{256025}(18902,\cdot)\)
\(\chi_{256025}(19023,\cdot)\)
\(\chi_{256025}(19567,\cdot)\)
\(\chi_{256025}(19842,\cdot)\)
\(\chi_{256025}(20183,\cdot)\)
\(\chi_{256025}(20738,\cdot)\)
\(\chi_{256025}(24033,\cdot)\)
\(\chi_{256025}(25237,\cdot)\)
\(\chi_{256025}(25503,\cdot)\)
\(\chi_{256025}(28163,\cdot)\)
\(\chi_{256025}(29353,\cdot)\)
\(\chi_{256025}(29397,\cdot)\)
...
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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 };
\((112652,198551,232751,67376)\) → \((e\left(\frac{13}{20}\right),e\left(\frac{11}{21}\right),e\left(\frac{3}{5}\right),e\left(\frac{7}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
| \( \chi_{ 256025 }(19842, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{163}{252}\right)\) | \(e\left(\frac{1241}{1260}\right)\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{199}{315}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{611}{630}\right)\) | \(e\left(\frac{39}{140}\right)\) | \(e\left(\frac{157}{1260}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{911}{1260}\right)\) |
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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)
