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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3888, base_ring=CyclotomicField(108))
M = H._module
chi = DirichletCharacter(H, M([0,27,32]))
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gp:[g,chi] = znchar(Mod(2773, 3888))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3888.2773");
| Modulus: | \(3888\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1296\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(108\) |
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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 \(\chi_{1296}(1141,\cdot)\) |
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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: | no |
| 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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\(\chi_{3888}(37,\cdot)\)
\(\chi_{3888}(181,\cdot)\)
\(\chi_{3888}(253,\cdot)\)
\(\chi_{3888}(397,\cdot)\)
\(\chi_{3888}(469,\cdot)\)
\(\chi_{3888}(613,\cdot)\)
\(\chi_{3888}(685,\cdot)\)
\(\chi_{3888}(829,\cdot)\)
\(\chi_{3888}(901,\cdot)\)
\(\chi_{3888}(1045,\cdot)\)
\(\chi_{3888}(1117,\cdot)\)
\(\chi_{3888}(1261,\cdot)\)
\(\chi_{3888}(1333,\cdot)\)
\(\chi_{3888}(1477,\cdot)\)
\(\chi_{3888}(1549,\cdot)\)
\(\chi_{3888}(1693,\cdot)\)
\(\chi_{3888}(1765,\cdot)\)
\(\chi_{3888}(1909,\cdot)\)
\(\chi_{3888}(1981,\cdot)\)
\(\chi_{3888}(2125,\cdot)\)
\(\chi_{3888}(2197,\cdot)\)
\(\chi_{3888}(2341,\cdot)\)
\(\chi_{3888}(2413,\cdot)\)
\(\chi_{3888}(2557,\cdot)\)
\(\chi_{3888}(2629,\cdot)\)
\(\chi_{3888}(2773,\cdot)\)
\(\chi_{3888}(2845,\cdot)\)
\(\chi_{3888}(2989,\cdot)\)
\(\chi_{3888}(3061,\cdot)\)
\(\chi_{3888}(3205,\cdot)\)
...
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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 };
\((2431,2917,1217)\) → \((1,i,e\left(\frac{8}{27}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 3888 }(2773, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{108}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{11}{108}\right)\) | \(e\left(\frac{13}{108}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{77}{108}\right)\) | \(e\left(\frac{25}{27}\right)\) |
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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)
