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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6175, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([63,165,140]))
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gp:[g,chi] = znchar(Mod(1203, 6175))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6175.1203");
| Modulus: | \(6175\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(6175\) |
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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);
|
\(\chi_{6175}(422,\cdot)\)
\(\chi_{6175}(453,\cdot)\)
\(\chi_{6175}(587,\cdot)\)
\(\chi_{6175}(617,\cdot)\)
\(\chi_{6175}(747,\cdot)\)
\(\chi_{6175}(878,\cdot)\)
\(\chi_{6175}(1042,\cdot)\)
\(\chi_{6175}(1073,\cdot)\)
\(\chi_{6175}(1203,\cdot)\)
\(\chi_{6175}(1233,\cdot)\)
\(\chi_{6175}(1688,\cdot)\)
\(\chi_{6175}(1753,\cdot)\)
\(\chi_{6175}(1822,\cdot)\)
\(\chi_{6175}(1852,\cdot)\)
\(\chi_{6175}(2113,\cdot)\)
\(\chi_{6175}(2277,\cdot)\)
\(\chi_{6175}(2308,\cdot)\)
\(\chi_{6175}(2342,\cdot)\)
\(\chi_{6175}(2438,\cdot)\)
\(\chi_{6175}(2892,\cdot)\)
\(\chi_{6175}(2923,\cdot)\)
\(\chi_{6175}(2988,\cdot)\)
\(\chi_{6175}(3087,\cdot)\)
\(\chi_{6175}(3217,\cdot)\)
\(\chi_{6175}(3348,\cdot)\)
\(\chi_{6175}(3512,\cdot)\)
\(\chi_{6175}(3577,\cdot)\)
\(\chi_{6175}(3673,\cdot)\)
\(\chi_{6175}(3703,\cdot)\)
\(\chi_{6175}(4127,\cdot)\)
...
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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 };
\((1977,951,3251)\) → \((e\left(\frac{7}{20}\right),e\left(\frac{11}{12}\right),e\left(\frac{7}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 6175 }(1203, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{41}{180}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{49}{180}\right)\) | \(-1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{41}{90}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{49}{90}\right)\) |
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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)
