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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3648, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([0,45,0,32]))
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gp:[g,chi] = znchar(Mod(757, 3648))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3648.757");
| Modulus: | \(3648\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1216\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(144\) |
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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 \(\chi_{1216}(757,\cdot)\) |
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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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\(\chi_{3648}(61,\cdot)\)
\(\chi_{3648}(85,\cdot)\)
\(\chi_{3648}(157,\cdot)\)
\(\chi_{3648}(253,\cdot)\)
\(\chi_{3648}(301,\cdot)\)
\(\chi_{3648}(397,\cdot)\)
\(\chi_{3648}(517,\cdot)\)
\(\chi_{3648}(541,\cdot)\)
\(\chi_{3648}(613,\cdot)\)
\(\chi_{3648}(709,\cdot)\)
\(\chi_{3648}(757,\cdot)\)
\(\chi_{3648}(853,\cdot)\)
\(\chi_{3648}(973,\cdot)\)
\(\chi_{3648}(997,\cdot)\)
\(\chi_{3648}(1069,\cdot)\)
\(\chi_{3648}(1165,\cdot)\)
\(\chi_{3648}(1213,\cdot)\)
\(\chi_{3648}(1309,\cdot)\)
\(\chi_{3648}(1429,\cdot)\)
\(\chi_{3648}(1453,\cdot)\)
\(\chi_{3648}(1525,\cdot)\)
\(\chi_{3648}(1621,\cdot)\)
\(\chi_{3648}(1669,\cdot)\)
\(\chi_{3648}(1765,\cdot)\)
\(\chi_{3648}(1885,\cdot)\)
\(\chi_{3648}(1909,\cdot)\)
\(\chi_{3648}(1981,\cdot)\)
\(\chi_{3648}(2077,\cdot)\)
\(\chi_{3648}(2125,\cdot)\)
\(\chi_{3648}(2221,\cdot)\)
...
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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 };
\((2623,2053,1217,1921)\) → \((1,e\left(\frac{5}{16}\right),1,e\left(\frac{2}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3648 }(757, a) \) |
\(1\) | \(1\) | \(e\left(\frac{125}{144}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{115}{144}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{31}{144}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{47}{144}\right)\) |
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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)
