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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2547, base_ring=CyclotomicField(282))
M = H._module
chi = DirichletCharacter(H, M([188,3]))
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gp:[g,chi] = znchar(Mod(1159, 2547))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2547.1159");
| Modulus: | \(2547\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2547\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(282\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{2547}(43,\cdot)\)
\(\chi_{2547}(58,\cdot)\)
\(\chi_{2547}(67,\cdot)\)
\(\chi_{2547}(76,\cdot)\)
\(\chi_{2547}(79,\cdot)\)
\(\chi_{2547}(115,\cdot)\)
\(\chi_{2547}(142,\cdot)\)
\(\chi_{2547}(205,\cdot)\)
\(\chi_{2547}(223,\cdot)\)
\(\chi_{2547}(229,\cdot)\)
\(\chi_{2547}(232,\cdot)\)
\(\chi_{2547}(241,\cdot)\)
\(\chi_{2547}(268,\cdot)\)
\(\chi_{2547}(304,\cdot)\)
\(\chi_{2547}(310,\cdot)\)
\(\chi_{2547}(313,\cdot)\)
\(\chi_{2547}(322,\cdot)\)
\(\chi_{2547}(367,\cdot)\)
\(\chi_{2547}(385,\cdot)\)
\(\chi_{2547}(391,\cdot)\)
\(\chi_{2547}(403,\cdot)\)
\(\chi_{2547}(439,\cdot)\)
\(\chi_{2547}(502,\cdot)\)
\(\chi_{2547}(562,\cdot)\)
\(\chi_{2547}(574,\cdot)\)
\(\chi_{2547}(598,\cdot)\)
\(\chi_{2547}(619,\cdot)\)
\(\chi_{2547}(688,\cdot)\)
\(\chi_{2547}(691,\cdot)\)
\(\chi_{2547}(697,\cdot)\)
...
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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 };
\((1982,1135)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{94}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2547 }(1159, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{101}{282}\right)\) | \(e\left(\frac{101}{141}\right)\) | \(e\left(\frac{229}{282}\right)\) | \(e\left(\frac{88}{141}\right)\) | \(e\left(\frac{7}{94}\right)\) | \(e\left(\frac{8}{47}\right)\) | \(e\left(\frac{40}{141}\right)\) | \(e\left(\frac{5}{141}\right)\) | \(e\left(\frac{277}{282}\right)\) | \(e\left(\frac{61}{141}\right)\) |
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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)
