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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8464, base_ring=CyclotomicField(1012))
M = H._module
chi = DirichletCharacter(H, M([0,759,738]))
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gp:[g,chi] = znchar(Mod(61, 8464))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8464.61");
| 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: | \(8464\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1012\) |
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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: | 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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\(\chi_{8464}(5,\cdot)\)
\(\chi_{8464}(21,\cdot)\)
\(\chi_{8464}(37,\cdot)\)
\(\chi_{8464}(53,\cdot)\)
\(\chi_{8464}(61,\cdot)\)
\(\chi_{8464}(109,\cdot)\)
\(\chi_{8464}(125,\cdot)\)
\(\chi_{8464}(149,\cdot)\)
\(\chi_{8464}(157,\cdot)\)
\(\chi_{8464}(181,\cdot)\)
\(\chi_{8464}(189,\cdot)\)
\(\chi_{8464}(205,\cdot)\)
\(\chi_{8464}(221,\cdot)\)
\(\chi_{8464}(237,\cdot)\)
\(\chi_{8464}(245,\cdot)\)
\(\chi_{8464}(293,\cdot)\)
\(\chi_{8464}(309,\cdot)\)
\(\chi_{8464}(333,\cdot)\)
\(\chi_{8464}(341,\cdot)\)
\(\chi_{8464}(365,\cdot)\)
\(\chi_{8464}(373,\cdot)\)
\(\chi_{8464}(389,\cdot)\)
\(\chi_{8464}(405,\cdot)\)
\(\chi_{8464}(421,\cdot)\)
\(\chi_{8464}(429,\cdot)\)
\(\chi_{8464}(477,\cdot)\)
\(\chi_{8464}(493,\cdot)\)
\(\chi_{8464}(517,\cdot)\)
\(\chi_{8464}(525,\cdot)\)
\(\chi_{8464}(549,\cdot)\)
...
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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: |
$\Q(\zeta_{1012})$ |
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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 1012 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((7407,2117,6353)\) → \((1,-i,e\left(\frac{369}{506}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 8464 }(61, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{929}{1012}\right)\) | \(e\left(\frac{485}{1012}\right)\) | \(e\left(\frac{145}{253}\right)\) | \(e\left(\frac{423}{506}\right)\) | \(e\left(\frac{185}{1012}\right)\) | \(e\left(\frac{817}{1012}\right)\) | \(e\left(\frac{201}{506}\right)\) | \(e\left(\frac{75}{506}\right)\) | \(e\left(\frac{147}{1012}\right)\) | \(e\left(\frac{497}{1012}\right)\) |
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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)
