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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(65869, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([130,198]))
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gp:[g,chi] = znchar(Mod(16542, 65869))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("65869.16542");
| Modulus: | \(65869\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(65869\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(165\) |
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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);
|
\(\chi_{65869}(64,\cdot)\)
\(\chi_{65869}(1779,\cdot)\)
\(\chi_{65869}(1978,\cdot)\)
\(\chi_{65869}(3129,\cdot)\)
\(\chi_{65869}(3705,\cdot)\)
\(\chi_{65869}(4096,\cdot)\)
\(\chi_{65869}(4295,\cdot)\)
\(\chi_{65869}(4784,\cdot)\)
\(\chi_{65869}(5115,\cdot)\)
\(\chi_{65869}(6022,\cdot)\)
\(\chi_{65869}(6684,\cdot)\)
\(\chi_{65869}(7346,\cdot)\)
\(\chi_{65869}(9087,\cdot)\)
\(\chi_{65869}(9663,\cdot)\)
\(\chi_{65869}(11980,\cdot)\)
\(\chi_{65869}(12311,\cdot)\)
\(\chi_{65869}(13033,\cdot)\)
\(\chi_{65869}(13232,\cdot)\)
\(\chi_{65869}(14714,\cdot)\)
\(\chi_{65869}(16343,\cdot)\)
\(\chi_{65869}(16542,\cdot)\)
\(\chi_{65869}(16674,\cdot)\)
\(\chi_{65869}(16873,\cdot)\)
\(\chi_{65869}(17005,\cdot)\)
\(\chi_{65869}(17031,\cdot)\)
\(\chi_{65869}(17204,\cdot)\)
\(\chi_{65869}(17938,\cdot)\)
\(\chi_{65869}(25968,\cdot)\)
\(\chi_{65869}(26213,\cdot)\)
\(\chi_{65869}(26273,\cdot)\)
...
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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 };
\((64877,996)\) → \((e\left(\frac{13}{33}\right),e\left(\frac{3}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 65869 }(16542, a) \) |
\(1\) | \(1\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{118}{165}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{58}{165}\right)\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{163}{165}\right)\) | \(e\left(\frac{53}{165}\right)\) | \(e\left(\frac{36}{55}\right)\) |
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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)
