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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5184, base_ring=CyclotomicField(432))
M = H._module
chi = DirichletCharacter(H, M([216,81,280]))
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gp:[g,chi] = znchar(Mod(1283, 5184))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5184.1283");
| Modulus: | \(5184\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5184\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(432\) |
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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: | yes |
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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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\(\chi_{5184}(11,\cdot)\)
\(\chi_{5184}(59,\cdot)\)
\(\chi_{5184}(83,\cdot)\)
\(\chi_{5184}(131,\cdot)\)
\(\chi_{5184}(155,\cdot)\)
\(\chi_{5184}(203,\cdot)\)
\(\chi_{5184}(227,\cdot)\)
\(\chi_{5184}(275,\cdot)\)
\(\chi_{5184}(299,\cdot)\)
\(\chi_{5184}(347,\cdot)\)
\(\chi_{5184}(371,\cdot)\)
\(\chi_{5184}(419,\cdot)\)
\(\chi_{5184}(443,\cdot)\)
\(\chi_{5184}(491,\cdot)\)
\(\chi_{5184}(515,\cdot)\)
\(\chi_{5184}(563,\cdot)\)
\(\chi_{5184}(587,\cdot)\)
\(\chi_{5184}(635,\cdot)\)
\(\chi_{5184}(659,\cdot)\)
\(\chi_{5184}(707,\cdot)\)
\(\chi_{5184}(731,\cdot)\)
\(\chi_{5184}(779,\cdot)\)
\(\chi_{5184}(803,\cdot)\)
\(\chi_{5184}(851,\cdot)\)
\(\chi_{5184}(875,\cdot)\)
\(\chi_{5184}(923,\cdot)\)
\(\chi_{5184}(947,\cdot)\)
\(\chi_{5184}(995,\cdot)\)
\(\chi_{5184}(1019,\cdot)\)
\(\chi_{5184}(1067,\cdot)\)
...
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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 };
| Field of values: |
$\Q(\zeta_{432})$ |
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sage:CyclotomicField(chi.multiplicative_order())
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gp:nfinit(polcyclo(charorder(g,chi)))
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magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 432 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((2431,325,1217)\) → \((-1,e\left(\frac{3}{16}\right),e\left(\frac{35}{54}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 5184 }(1283, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{432}\right)\) | \(e\left(\frac{161}{216}\right)\) | \(e\left(\frac{373}{432}\right)\) | \(e\left(\frac{431}{432}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{133}{144}\right)\) | \(e\left(\frac{55}{216}\right)\) | \(e\left(\frac{41}{216}\right)\) | \(e\left(\frac{19}{432}\right)\) | \(e\left(\frac{26}{27}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
