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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4232, base_ring=CyclotomicField(506))
M = H._module
chi = DirichletCharacter(H, M([0,0,191]))
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gp:[g,chi] = znchar(Mod(65, 4232))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4232.65");
| Modulus: | \(4232\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(529\) |
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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 \(\chi_{529}(65,\cdot)\) |
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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: | yes |
| 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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\(\chi_{4232}(17,\cdot)\)
\(\chi_{4232}(33,\cdot)\)
\(\chi_{4232}(57,\cdot)\)
\(\chi_{4232}(65,\cdot)\)
\(\chi_{4232}(89,\cdot)\)
\(\chi_{4232}(97,\cdot)\)
\(\chi_{4232}(113,\cdot)\)
\(\chi_{4232}(129,\cdot)\)
\(\chi_{4232}(145,\cdot)\)
\(\chi_{4232}(153,\cdot)\)
\(\chi_{4232}(201,\cdot)\)
\(\chi_{4232}(217,\cdot)\)
\(\chi_{4232}(241,\cdot)\)
\(\chi_{4232}(249,\cdot)\)
\(\chi_{4232}(273,\cdot)\)
\(\chi_{4232}(281,\cdot)\)
\(\chi_{4232}(297,\cdot)\)
\(\chi_{4232}(313,\cdot)\)
\(\chi_{4232}(329,\cdot)\)
\(\chi_{4232}(337,\cdot)\)
\(\chi_{4232}(385,\cdot)\)
\(\chi_{4232}(401,\cdot)\)
\(\chi_{4232}(425,\cdot)\)
\(\chi_{4232}(433,\cdot)\)
\(\chi_{4232}(457,\cdot)\)
\(\chi_{4232}(465,\cdot)\)
\(\chi_{4232}(481,\cdot)\)
\(\chi_{4232}(497,\cdot)\)
\(\chi_{4232}(513,\cdot)\)
\(\chi_{4232}(521,\cdot)\)
...
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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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\((3175,2117,2121)\) → \((1,1,e\left(\frac{191}{506}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 4232 }(65, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{10}{253}\right)\) | \(e\left(\frac{191}{506}\right)\) | \(e\left(\frac{351}{506}\right)\) | \(e\left(\frac{20}{253}\right)\) | \(e\left(\frac{245}{506}\right)\) | \(e\left(\frac{182}{253}\right)\) | \(e\left(\frac{211}{506}\right)\) | \(e\left(\frac{479}{506}\right)\) | \(e\left(\frac{181}{506}\right)\) | \(e\left(\frac{371}{506}\right)\) |
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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)
