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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6010, base_ring=CyclotomicField(200))
M = H._module
chi = DirichletCharacter(H, M([0,73]))
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gp:[g,chi] = znchar(Mod(181, 6010))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6010.181");
| Modulus: | \(6010\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(601\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(200\) |
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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: | no, induced from \(\chi_{601}(181,\cdot)\) |
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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);
|
\(\chi_{6010}(31,\cdot)\)
\(\chi_{6010}(51,\cdot)\)
\(\chi_{6010}(171,\cdot)\)
\(\chi_{6010}(181,\cdot)\)
\(\chi_{6010}(191,\cdot)\)
\(\chi_{6010}(251,\cdot)\)
\(\chi_{6010}(261,\cdot)\)
\(\chi_{6010}(271,\cdot)\)
\(\chi_{6010}(391,\cdot)\)
\(\chi_{6010}(431,\cdot)\)
\(\chi_{6010}(771,\cdot)\)
\(\chi_{6010}(811,\cdot)\)
\(\chi_{6010}(931,\cdot)\)
\(\chi_{6010}(941,\cdot)\)
\(\chi_{6010}(951,\cdot)\)
\(\chi_{6010}(1011,\cdot)\)
\(\chi_{6010}(1021,\cdot)\)
\(\chi_{6010}(1031,\cdot)\)
\(\chi_{6010}(1151,\cdot)\)
\(\chi_{6010}(1171,\cdot)\)
\(\chi_{6010}(1281,\cdot)\)
\(\chi_{6010}(1311,\cdot)\)
\(\chi_{6010}(1331,\cdot)\)
\(\chi_{6010}(1421,\cdot)\)
\(\chi_{6010}(1441,\cdot)\)
\(\chi_{6010}(1461,\cdot)\)
\(\chi_{6010}(1471,\cdot)\)
\(\chi_{6010}(1551,\cdot)\)
\(\chi_{6010}(1701,\cdot)\)
\(\chi_{6010}(1741,\cdot)\)
...
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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 };
\((3607,2411)\) → \((1,e\left(\frac{73}{200}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 6010 }(181, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{73}{200}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{61}{200}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{193}{200}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{23}{100}\right)\) | \(e\left(\frac{22}{25}\right)\) |
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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)
