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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6004, base_ring=CyclotomicField(234))
M = H._module
chi = DirichletCharacter(H, M([0,91,147]))
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gp:[g,chi] = znchar(Mod(1781, 6004))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6004.1781");
| Modulus: | \(6004\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1501\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(234\) |
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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 \(\chi_{1501}(280,\cdot)\) |
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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);
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\(\chi_{6004}(29,\cdot)\)
\(\chi_{6004}(53,\cdot)\)
\(\chi_{6004}(109,\cdot)\)
\(\chi_{6004}(165,\cdot)\)
\(\chi_{6004}(193,\cdot)\)
\(\chi_{6004}(205,\cdot)\)
\(\chi_{6004}(469,\cdot)\)
\(\chi_{6004}(477,\cdot)\)
\(\chi_{6004}(621,\cdot)\)
\(\chi_{6004}(661,\cdot)\)
\(\chi_{6004}(793,\cdot)\)
\(\chi_{6004}(865,\cdot)\)
\(\chi_{6004}(1001,\cdot)\)
\(\chi_{6004}(1153,\cdot)\)
\(\chi_{6004}(1245,\cdot)\)
\(\chi_{6004}(1409,\cdot)\)
\(\chi_{6004}(1465,\cdot)\)
\(\chi_{6004}(1485,\cdot)\)
\(\chi_{6004}(1497,\cdot)\)
\(\chi_{6004}(1549,\cdot)\)
\(\chi_{6004}(1609,\cdot)\)
\(\chi_{6004}(1617,\cdot)\)
\(\chi_{6004}(1693,\cdot)\)
\(\chi_{6004}(1713,\cdot)\)
\(\chi_{6004}(1777,\cdot)\)
\(\chi_{6004}(1781,\cdot)\)
\(\chi_{6004}(1845,\cdot)\)
\(\chi_{6004}(1877,\cdot)\)
\(\chi_{6004}(1933,\cdot)\)
\(\chi_{6004}(2009,\cdot)\)
...
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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 };
\((3003,2529,3953)\) → \((1,e\left(\frac{7}{18}\right),e\left(\frac{49}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 6004 }(1781, a) \) |
\(1\) | \(1\) | \(e\left(\frac{80}{117}\right)\) | \(e\left(\frac{20}{117}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{43}{117}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{71}{234}\right)\) | \(e\left(\frac{100}{117}\right)\) | \(e\left(\frac{19}{234}\right)\) | \(e\left(\frac{73}{234}\right)\) | \(e\left(\frac{1}{9}\right)\) |
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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)
