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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(22218, base_ring=CyclotomicField(1518))
M = H._module
chi = DirichletCharacter(H, M([0,1265,1218]))
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gp:[g,chi] = znchar(Mod(607, 22218))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("22218.607");
| Modulus: | \(22218\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3703\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1518\) |
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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: | no, induced from \(\chi_{3703}(607,\cdot)\) |
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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);
|
\(\chi_{22218}(31,\cdot)\)
\(\chi_{22218}(73,\cdot)\)
\(\chi_{22218}(187,\cdot)\)
\(\chi_{22218}(271,\cdot)\)
\(\chi_{22218}(325,\cdot)\)
\(\chi_{22218}(397,\cdot)\)
\(\chi_{22218}(409,\cdot)\)
\(\chi_{22218}(439,\cdot)\)
\(\chi_{22218}(535,\cdot)\)
\(\chi_{22218}(565,\cdot)\)
\(\chi_{22218}(577,\cdot)\)
\(\chi_{22218}(607,\cdot)\)
\(\chi_{22218}(703,\cdot)\)
\(\chi_{22218}(745,\cdot)\)
\(\chi_{22218}(775,\cdot)\)
\(\chi_{22218}(817,\cdot)\)
\(\chi_{22218}(859,\cdot)\)
\(\chi_{22218}(901,\cdot)\)
\(\chi_{22218}(913,\cdot)\)
\(\chi_{22218}(955,\cdot)\)
\(\chi_{22218}(997,\cdot)\)
\(\chi_{22218}(1039,\cdot)\)
\(\chi_{22218}(1153,\cdot)\)
\(\chi_{22218}(1237,\cdot)\)
\(\chi_{22218}(1291,\cdot)\)
\(\chi_{22218}(1363,\cdot)\)
\(\chi_{22218}(1375,\cdot)\)
\(\chi_{22218}(1405,\cdot)\)
\(\chi_{22218}(1501,\cdot)\)
\(\chi_{22218}(1531,\cdot)\)
...
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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 };
\((14813,9523,10585)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{203}{253}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 22218 }(607, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1471}{1518}\right)\) | \(e\left(\frac{454}{759}\right)\) | \(e\left(\frac{481}{506}\right)\) | \(e\left(\frac{155}{1518}\right)\) | \(e\left(\frac{835}{1518}\right)\) | \(e\left(\frac{712}{759}\right)\) | \(e\left(\frac{90}{253}\right)\) | \(e\left(\frac{59}{1518}\right)\) | \(e\left(\frac{722}{759}\right)\) | \(e\left(\frac{505}{506}\right)\) |
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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)
