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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2869, base_ring=CyclotomicField(450))
M = H._module
chi = DirichletCharacter(H, M([325,369]))
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gp:[g,chi] = znchar(Mod(687, 2869))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2869.687");
| Modulus: | \(2869\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2869\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(450\) |
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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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\(\chi_{2869}(3,\cdot)\)
\(\chi_{2869}(41,\cdot)\)
\(\chi_{2869}(53,\cdot)\)
\(\chi_{2869}(60,\cdot)\)
\(\chi_{2869}(67,\cdot)\)
\(\chi_{2869}(70,\cdot)\)
\(\chi_{2869}(79,\cdot)\)
\(\chi_{2869}(154,\cdot)\)
\(\chi_{2869}(192,\cdot)\)
\(\chi_{2869}(204,\cdot)\)
\(\chi_{2869}(211,\cdot)\)
\(\chi_{2869}(224,\cdot)\)
\(\chi_{2869}(230,\cdot)\)
\(\chi_{2869}(326,\cdot)\)
\(\chi_{2869}(355,\cdot)\)
\(\chi_{2869}(375,\cdot)\)
\(\chi_{2869}(409,\cdot)\)
\(\chi_{2869}(433,\cdot)\)
\(\chi_{2869}(477,\cdot)\)
\(\chi_{2869}(523,\cdot)\)
\(\chi_{2869}(526,\cdot)\)
\(\chi_{2869}(554,\cdot)\)
\(\chi_{2869}(584,\cdot)\)
\(\chi_{2869}(630,\cdot)\)
\(\chi_{2869}(661,\cdot)\)
\(\chi_{2869}(687,\cdot)\)
\(\chi_{2869}(705,\cdot)\)
\(\chi_{2869}(735,\cdot)\)
\(\chi_{2869}(781,\cdot)\)
\(\chi_{2869}(782,\cdot)\)
...
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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 };
\((2719,761)\) → \((e\left(\frac{13}{18}\right),e\left(\frac{41}{50}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 2869 }(687, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{182}{225}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{89}{225}\right)\) | \(e\left(\frac{419}{450}\right)\) | \(e\left(\frac{41}{150}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{139}{225}\right)\) | \(e\left(\frac{233}{450}\right)\) | \(e\left(\frac{56}{75}\right)\) |
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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)
