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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2592, base_ring=CyclotomicField(216))
M = H._module
chi = DirichletCharacter(H, M([0,135,8]))
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gp:[g,chi] = znchar(Mod(85, 2592))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2592.85");
| Modulus: | \(2592\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
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| Conductor: | \(2592\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(216\) |
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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;
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| 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);
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| 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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\(\chi_{2592}(13,\cdot)\)
\(\chi_{2592}(61,\cdot)\)
\(\chi_{2592}(85,\cdot)\)
\(\chi_{2592}(133,\cdot)\)
\(\chi_{2592}(157,\cdot)\)
\(\chi_{2592}(205,\cdot)\)
\(\chi_{2592}(229,\cdot)\)
\(\chi_{2592}(277,\cdot)\)
\(\chi_{2592}(301,\cdot)\)
\(\chi_{2592}(349,\cdot)\)
\(\chi_{2592}(373,\cdot)\)
\(\chi_{2592}(421,\cdot)\)
\(\chi_{2592}(445,\cdot)\)
\(\chi_{2592}(493,\cdot)\)
\(\chi_{2592}(517,\cdot)\)
\(\chi_{2592}(565,\cdot)\)
\(\chi_{2592}(589,\cdot)\)
\(\chi_{2592}(637,\cdot)\)
\(\chi_{2592}(661,\cdot)\)
\(\chi_{2592}(709,\cdot)\)
\(\chi_{2592}(733,\cdot)\)
\(\chi_{2592}(781,\cdot)\)
\(\chi_{2592}(805,\cdot)\)
\(\chi_{2592}(853,\cdot)\)
\(\chi_{2592}(877,\cdot)\)
\(\chi_{2592}(925,\cdot)\)
\(\chi_{2592}(949,\cdot)\)
\(\chi_{2592}(997,\cdot)\)
\(\chi_{2592}(1021,\cdot)\)
\(\chi_{2592}(1069,\cdot)\)
...
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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: |
$\Q(\zeta_{216})$ |
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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));
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| Fixed field: |
Number field defined by a degree 216 polynomial (not computed) |
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sage:chi.fixed_field()
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\((2431,325,1217)\) → \((1,e\left(\frac{5}{8}\right),e\left(\frac{1}{27}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 2592 }(85, a) \) |
\(1\) | \(1\) | \(e\left(\frac{103}{216}\right)\) | \(e\left(\frac{91}{108}\right)\) | \(e\left(\frac{131}{216}\right)\) | \(e\left(\frac{145}{216}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{17}{108}\right)\) | \(e\left(\frac{103}{108}\right)\) | \(e\left(\frac{53}{216}\right)\) | \(e\left(\frac{20}{27}\right)\) |
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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)
