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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2065, base_ring=CyclotomicField(174))
M = H._module
chi = DirichletCharacter(H, M([87,145,3]))
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gp:[g,chi] = znchar(Mod(474, 2065))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2065.474");
| Modulus: | \(2065\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2065\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(174\) |
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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);
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\(\chi_{2065}(24,\cdot)\)
\(\chi_{2065}(54,\cdot)\)
\(\chi_{2065}(89,\cdot)\)
\(\chi_{2065}(124,\cdot)\)
\(\chi_{2065}(129,\cdot)\)
\(\chi_{2065}(229,\cdot)\)
\(\chi_{2065}(269,\cdot)\)
\(\chi_{2065}(334,\cdot)\)
\(\chi_{2065}(339,\cdot)\)
\(\chi_{2065}(404,\cdot)\)
\(\chi_{2065}(409,\cdot)\)
\(\chi_{2065}(444,\cdot)\)
\(\chi_{2065}(474,\cdot)\)
\(\chi_{2065}(509,\cdot)\)
\(\chi_{2065}(514,\cdot)\)
\(\chi_{2065}(544,\cdot)\)
\(\chi_{2065}(549,\cdot)\)
\(\chi_{2065}(614,\cdot)\)
\(\chi_{2065}(689,\cdot)\)
\(\chi_{2065}(719,\cdot)\)
\(\chi_{2065}(859,\cdot)\)
\(\chi_{2065}(864,\cdot)\)
\(\chi_{2065}(899,\cdot)\)
\(\chi_{2065}(929,\cdot)\)
\(\chi_{2065}(999,\cdot)\)
\(\chi_{2065}(1034,\cdot)\)
\(\chi_{2065}(1104,\cdot)\)
\(\chi_{2065}(1109,\cdot)\)
\(\chi_{2065}(1139,\cdot)\)
\(\chi_{2065}(1144,\cdot)\)
...
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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 };
\((827,591,946)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{1}{58}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 2065 }(474, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{87}\right)\) | \(e\left(\frac{17}{87}\right)\) | \(e\left(\frac{32}{87}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{34}{87}\right)\) | \(e\left(\frac{133}{174}\right)\) | \(e\left(\frac{49}{87}\right)\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{64}{87}\right)\) |
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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)
