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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1323, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([35,39]))
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gp:[g,chi] = znchar(Mod(59, 1323))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1323.59");
| Modulus: | \(1323\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1323\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(126\) |
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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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\(\chi_{1323}(47,\cdot)\)
\(\chi_{1323}(59,\cdot)\)
\(\chi_{1323}(110,\cdot)\)
\(\chi_{1323}(122,\cdot)\)
\(\chi_{1323}(173,\cdot)\)
\(\chi_{1323}(185,\cdot)\)
\(\chi_{1323}(236,\cdot)\)
\(\chi_{1323}(248,\cdot)\)
\(\chi_{1323}(299,\cdot)\)
\(\chi_{1323}(311,\cdot)\)
\(\chi_{1323}(425,\cdot)\)
\(\chi_{1323}(437,\cdot)\)
\(\chi_{1323}(488,\cdot)\)
\(\chi_{1323}(500,\cdot)\)
\(\chi_{1323}(551,\cdot)\)
\(\chi_{1323}(563,\cdot)\)
\(\chi_{1323}(614,\cdot)\)
\(\chi_{1323}(626,\cdot)\)
\(\chi_{1323}(677,\cdot)\)
\(\chi_{1323}(689,\cdot)\)
\(\chi_{1323}(740,\cdot)\)
\(\chi_{1323}(752,\cdot)\)
\(\chi_{1323}(866,\cdot)\)
\(\chi_{1323}(878,\cdot)\)
\(\chi_{1323}(929,\cdot)\)
\(\chi_{1323}(941,\cdot)\)
\(\chi_{1323}(992,\cdot)\)
\(\chi_{1323}(1004,\cdot)\)
\(\chi_{1323}(1055,\cdot)\)
\(\chi_{1323}(1067,\cdot)\)
...
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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: |
$\Q(\zeta_{63})$ |
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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 126 polynomial (not computed) |
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sage:chi.fixed_field()
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\((785,1081)\) → \((e\left(\frac{5}{18}\right),e\left(\frac{13}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1323 }(59, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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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)
