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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6888, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([60,60,60,100,9]))
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gp:[g,chi] = znchar(Mod(1979, 6888))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6888.1979");
| Modulus: | \(6888\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(6888\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(120\) |
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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);
|
\(\chi_{6888}(227,\cdot)\)
\(\chi_{6888}(299,\cdot)\)
\(\chi_{6888}(395,\cdot)\)
\(\chi_{6888}(563,\cdot)\)
\(\chi_{6888}(731,\cdot)\)
\(\chi_{6888}(803,\cdot)\)
\(\chi_{6888}(971,\cdot)\)
\(\chi_{6888}(1571,\cdot)\)
\(\chi_{6888}(1739,\cdot)\)
\(\chi_{6888}(1811,\cdot)\)
\(\chi_{6888}(1979,\cdot)\)
\(\chi_{6888}(2147,\cdot)\)
\(\chi_{6888}(2243,\cdot)\)
\(\chi_{6888}(2315,\cdot)\)
\(\chi_{6888}(2987,\cdot)\)
\(\chi_{6888}(3251,\cdot)\)
\(\chi_{6888}(3491,\cdot)\)
\(\chi_{6888}(3755,\cdot)\)
\(\chi_{6888}(3923,\cdot)\)
\(\chi_{6888}(4163,\cdot)\)
\(\chi_{6888}(4331,\cdot)\)
\(\chi_{6888}(4499,\cdot)\)
\(\chi_{6888}(4667,\cdot)\)
\(\chi_{6888}(4763,\cdot)\)
\(\chi_{6888}(4931,\cdot)\)
\(\chi_{6888}(5099,\cdot)\)
\(\chi_{6888}(5267,\cdot)\)
\(\chi_{6888}(5507,\cdot)\)
\(\chi_{6888}(5675,\cdot)\)
\(\chi_{6888}(5939,\cdot)\)
...
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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 };
\((5167,3445,2297,3937,5377)\) → \((-1,-1,-1,e\left(\frac{5}{6}\right),e\left(\frac{3}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 6888 }(1979, a) \) |
\(1\) | \(1\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) |
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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)
