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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1011, base_ring=CyclotomicField(336))
M = H._module
chi = DirichletCharacter(H, M([168,257]))
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gp:[g,chi] = znchar(Mod(29, 1011))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1011.29");
| Modulus: | \(1011\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1011\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(336\) |
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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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\(\chi_{1011}(20,\cdot)\)
\(\chi_{1011}(23,\cdot)\)
\(\chi_{1011}(29,\cdot)\)
\(\chi_{1011}(44,\cdot)\)
\(\chi_{1011}(53,\cdot)\)
\(\chi_{1011}(68,\cdot)\)
\(\chi_{1011}(71,\cdot)\)
\(\chi_{1011}(80,\cdot)\)
\(\chi_{1011}(83,\cdot)\)
\(\chi_{1011}(89,\cdot)\)
\(\chi_{1011}(101,\cdot)\)
\(\chi_{1011}(116,\cdot)\)
\(\chi_{1011}(134,\cdot)\)
\(\chi_{1011}(143,\cdot)\)
\(\chi_{1011}(152,\cdot)\)
\(\chi_{1011}(161,\cdot)\)
\(\chi_{1011}(176,\cdot)\)
\(\chi_{1011}(185,\cdot)\)
\(\chi_{1011}(194,\cdot)\)
\(\chi_{1011}(203,\cdot)\)
\(\chi_{1011}(221,\cdot)\)
\(\chi_{1011}(236,\cdot)\)
\(\chi_{1011}(248,\cdot)\)
\(\chi_{1011}(254,\cdot)\)
\(\chi_{1011}(257,\cdot)\)
\(\chi_{1011}(266,\cdot)\)
\(\chi_{1011}(269,\cdot)\)
\(\chi_{1011}(284,\cdot)\)
\(\chi_{1011}(293,\cdot)\)
\(\chi_{1011}(308,\cdot)\)
...
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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: |
$\Q(\zeta_{336})$ |
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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 336 polynomial (not computed) |
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sage:chi.fixed_field()
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\((338,10)\) → \((-1,e\left(\frac{257}{336}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1011 }(29, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{83}{112}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{257}{336}\right)\) | \(e\left(\frac{33}{112}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{43}{168}\right)\) | \(e\left(\frac{2}{21}\right)\) |
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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)
