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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4729, base_ring=CyclotomicField(788))
M = H._module
chi = DirichletCharacter(H, M([401]))
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gp:[g,chi] = znchar(Mod(335, 4729))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4729.335");
| Modulus: | \(4729\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4729\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(788\) |
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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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\(\chi_{4729}(2,\cdot)\)
\(\chi_{4729}(3,\cdot)\)
\(\chi_{4729}(8,\cdot)\)
\(\chi_{4729}(12,\cdot)\)
\(\chi_{4729}(18,\cdot)\)
\(\chi_{4729}(27,\cdot)\)
\(\chi_{4729}(32,\cdot)\)
\(\chi_{4729}(48,\cdot)\)
\(\chi_{4729}(72,\cdot)\)
\(\chi_{4729}(108,\cdot)\)
\(\chi_{4729}(121,\cdot)\)
\(\chi_{4729}(125,\cdot)\)
\(\chi_{4729}(127,\cdot)\)
\(\chi_{4729}(128,\cdot)\)
\(\chi_{4729}(157,\cdot)\)
\(\chi_{4729}(162,\cdot)\)
\(\chi_{4729}(175,\cdot)\)
\(\chi_{4729}(192,\cdot)\)
\(\chi_{4729}(211,\cdot)\)
\(\chi_{4729}(227,\cdot)\)
\(\chi_{4729}(229,\cdot)\)
\(\chi_{4729}(243,\cdot)\)
\(\chi_{4729}(245,\cdot)\)
\(\chi_{4729}(288,\cdot)\)
\(\chi_{4729}(317,\cdot)\)
\(\chi_{4729}(335,\cdot)\)
\(\chi_{4729}(341,\cdot)\)
\(\chi_{4729}(343,\cdot)\)
\(\chi_{4729}(367,\cdot)\)
\(\chi_{4729}(373,\cdot)\)
...
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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: |
$\Q(\zeta_{788})$ |
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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 788 polynomial (not computed) |
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sage:chi.fixed_field()
|
\(17\) → \(e\left(\frac{401}{788}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4729 }(335, a) \) |
\(1\) | \(1\) | \(e\left(\frac{325}{394}\right)\) | \(e\left(\frac{201}{394}\right)\) | \(e\left(\frac{128}{197}\right)\) | \(e\left(\frac{123}{394}\right)\) | \(e\left(\frac{66}{197}\right)\) | \(e\left(\frac{195}{394}\right)\) | \(e\left(\frac{187}{394}\right)\) | \(e\left(\frac{4}{197}\right)\) | \(e\left(\frac{27}{197}\right)\) | \(e\left(\frac{175}{788}\right)\) |
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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)
