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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1183, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([78,103]))
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gp:[g,chi] = znchar(Mod(349, 1183))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1183.349");
| Modulus: | \(1183\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1183\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(156\) |
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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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\(\chi_{1183}(6,\cdot)\)
\(\chi_{1183}(20,\cdot)\)
\(\chi_{1183}(41,\cdot)\)
\(\chi_{1183}(76,\cdot)\)
\(\chi_{1183}(97,\cdot)\)
\(\chi_{1183}(111,\cdot)\)
\(\chi_{1183}(132,\cdot)\)
\(\chi_{1183}(167,\cdot)\)
\(\chi_{1183}(202,\cdot)\)
\(\chi_{1183}(223,\cdot)\)
\(\chi_{1183}(279,\cdot)\)
\(\chi_{1183}(293,\cdot)\)
\(\chi_{1183}(314,\cdot)\)
\(\chi_{1183}(349,\cdot)\)
\(\chi_{1183}(370,\cdot)\)
\(\chi_{1183}(384,\cdot)\)
\(\chi_{1183}(405,\cdot)\)
\(\chi_{1183}(440,\cdot)\)
\(\chi_{1183}(461,\cdot)\)
\(\chi_{1183}(475,\cdot)\)
\(\chi_{1183}(496,\cdot)\)
\(\chi_{1183}(531,\cdot)\)
\(\chi_{1183}(552,\cdot)\)
\(\chi_{1183}(566,\cdot)\)
\(\chi_{1183}(622,\cdot)\)
\(\chi_{1183}(643,\cdot)\)
\(\chi_{1183}(678,\cdot)\)
\(\chi_{1183}(713,\cdot)\)
\(\chi_{1183}(734,\cdot)\)
\(\chi_{1183}(748,\cdot)\)
...
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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: |
$\Q(\zeta_{156})$ |
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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));
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| Fixed field: |
Number field defined by a degree 156 polynomial (not computed) |
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sage:chi.fixed_field()
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\((339,1016)\) → \((-1,e\left(\frac{103}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 1183 }(349, a) \) |
\(1\) | \(1\) | \(e\left(\frac{103}{156}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{5}{156}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{1}{156}\right)\) | \(e\left(\frac{9}{13}\right)\) |
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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)
