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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8464, base_ring=CyclotomicField(506))
M = H._module
chi = DirichletCharacter(H, M([253,253,166]))
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gp:[g,chi] = znchar(Mod(87, 8464))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8464.87");
| Modulus: | \(8464\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4232\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(506\) |
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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 \(\chi_{4232}(2203,\cdot)\) |
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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: | odd |
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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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\(\chi_{8464}(39,\cdot)\)
\(\chi_{8464}(55,\cdot)\)
\(\chi_{8464}(71,\cdot)\)
\(\chi_{8464}(87,\cdot)\)
\(\chi_{8464}(119,\cdot)\)
\(\chi_{8464}(151,\cdot)\)
\(\chi_{8464}(167,\cdot)\)
\(\chi_{8464}(215,\cdot)\)
\(\chi_{8464}(279,\cdot)\)
\(\chi_{8464}(311,\cdot)\)
\(\chi_{8464}(407,\cdot)\)
\(\chi_{8464}(423,\cdot)\)
\(\chi_{8464}(439,\cdot)\)
\(\chi_{8464}(455,\cdot)\)
\(\chi_{8464}(519,\cdot)\)
\(\chi_{8464}(535,\cdot)\)
\(\chi_{8464}(583,\cdot)\)
\(\chi_{8464}(679,\cdot)\)
\(\chi_{8464}(775,\cdot)\)
\(\chi_{8464}(791,\cdot)\)
\(\chi_{8464}(807,\cdot)\)
\(\chi_{8464}(823,\cdot)\)
\(\chi_{8464}(855,\cdot)\)
\(\chi_{8464}(887,\cdot)\)
\(\chi_{8464}(903,\cdot)\)
\(\chi_{8464}(951,\cdot)\)
\(\chi_{8464}(1015,\cdot)\)
\(\chi_{8464}(1047,\cdot)\)
\(\chi_{8464}(1143,\cdot)\)
\(\chi_{8464}(1159,\cdot)\)
...
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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: |
$\Q(\zeta_{253})$ |
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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 506 polynomial (not computed) |
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sage:chi.fixed_field()
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\((7407,2117,6353)\) → \((-1,-1,e\left(\frac{83}{253}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 8464 }(87, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{63}{253}\right)\) | \(e\left(\frac{419}{506}\right)\) | \(e\left(\frac{415}{506}\right)\) | \(e\left(\frac{126}{253}\right)\) | \(e\left(\frac{76}{253}\right)\) | \(e\left(\frac{421}{506}\right)\) | \(e\left(\frac{39}{506}\right)\) | \(e\left(\frac{130}{253}\right)\) | \(e\left(\frac{178}{253}\right)\) | \(e\left(\frac{35}{506}\right)\) |
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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)
