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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(24102, base_ring=CyclotomicField(204))
M = H._module
chi = DirichletCharacter(H, M([136,17,74]))
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gp:[g,chi] = znchar(Mod(6073, 24102))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("24102.6073");
| Modulus: | \(24102\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(12051\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(204\) |
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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 \(\chi_{12051}(6073,\cdot)\) |
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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: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{24102}(85,\cdot)\)
\(\chi_{24102}(115,\cdot)\)
\(\chi_{24102}(349,\cdot)\)
\(\chi_{24102}(457,\cdot)\)
\(\chi_{24102}(817,\cdot)\)
\(\chi_{24102}(1051,\cdot)\)
\(\chi_{24102}(1345,\cdot)\)
\(\chi_{24102}(1723,\cdot)\)
\(\chi_{24102}(1813,\cdot)\)
\(\chi_{24102}(2455,\cdot)\)
\(\chi_{24102}(2515,\cdot)\)
\(\chi_{24102}(2659,\cdot)\)
\(\chi_{24102}(2689,\cdot)\)
\(\chi_{24102}(3157,\cdot)\)
\(\chi_{24102}(3967,\cdot)\)
\(\chi_{24102}(4297,\cdot)\)
\(\chi_{24102}(5839,\cdot)\)
\(\chi_{24102}(6073,\cdot)\)
\(\chi_{24102}(7009,\cdot)\)
\(\chi_{24102}(7105,\cdot)\)
\(\chi_{24102}(7573,\cdot)\)
\(\chi_{24102}(8275,\cdot)\)
\(\chi_{24102}(8305,\cdot)\)
\(\chi_{24102}(8413,\cdot)\)
\(\chi_{24102}(8833,\cdot)\)
\(\chi_{24102}(9115,\cdot)\)
\(\chi_{24102}(9211,\cdot)\)
\(\chi_{24102}(9769,\cdot)\)
\(\chi_{24102}(10285,\cdot)\)
\(\chi_{24102}(10615,\cdot)\)
...
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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 };
\((5357,9271,13807)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{12}\right),e\left(\frac{37}{102}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 24102 }(6073, a) \) |
\(1\) | \(1\) | \(e\left(\frac{91}{204}\right)\) | \(e\left(\frac{7}{204}\right)\) | \(e\left(\frac{77}{204}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{89}{204}\right)\) | \(e\left(\frac{89}{102}\right)\) | \(e\left(\frac{91}{102}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{155}{204}\right)\) | \(e\left(\frac{49}{102}\right)\) |
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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)
