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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8456, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([75,75,50,29]))
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gp:[g,chi] = znchar(Mod(4867, 8456))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8456.4867");
| Modulus: | \(8456\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8456\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(150\) |
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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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\(\chi_{8456}(163,\cdot)\)
\(\chi_{8456}(291,\cdot)\)
\(\chi_{8456}(443,\cdot)\)
\(\chi_{8456}(459,\cdot)\)
\(\chi_{8456}(555,\cdot)\)
\(\chi_{8456}(1187,\cdot)\)
\(\chi_{8456}(1411,\cdot)\)
\(\chi_{8456}(1619,\cdot)\)
\(\chi_{8456}(1675,\cdot)\)
\(\chi_{8456}(1787,\cdot)\)
\(\chi_{8456}(2067,\cdot)\)
\(\chi_{8456}(2083,\cdot)\)
\(\chi_{8456}(2795,\cdot)\)
\(\chi_{8456}(2811,\cdot)\)
\(\chi_{8456}(2923,\cdot)\)
\(\chi_{8456}(3035,\cdot)\)
\(\chi_{8456}(3411,\cdot)\)
\(\chi_{8456}(4211,\cdot)\)
\(\chi_{8456}(4435,\cdot)\)
\(\chi_{8456}(4475,\cdot)\)
\(\chi_{8456}(4827,\cdot)\)
\(\chi_{8456}(4867,\cdot)\)
\(\chi_{8456}(4883,\cdot)\)
\(\chi_{8456}(5091,\cdot)\)
\(\chi_{8456}(5147,\cdot)\)
\(\chi_{8456}(5315,\cdot)\)
\(\chi_{8456}(5443,\cdot)\)
\(\chi_{8456}(5499,\cdot)\)
\(\chi_{8456}(6003,\cdot)\)
\(\chi_{8456}(6155,\cdot)\)
...
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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 };
\((6343,4229,4833,8009)\) → \((-1,-1,e\left(\frac{1}{3}\right),e\left(\frac{29}{150}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 8456 }(4867, a) \) |
\(1\) | \(1\) | \(e\left(\frac{149}{150}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{74}{75}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{32}{75}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{16}{25}\right)\) |
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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)
