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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2366, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([78,29]))
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gp:[g,chi] = znchar(Mod(1007, 2366))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2366.1007");
| Modulus: | \(2366\) |
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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: | no, induced from \(\chi_{1183}(1007,\cdot)\) |
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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: | 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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\(\chi_{2366}(41,\cdot)\)
\(\chi_{2366}(97,\cdot)\)
\(\chi_{2366}(111,\cdot)\)
\(\chi_{2366}(167,\cdot)\)
\(\chi_{2366}(223,\cdot)\)
\(\chi_{2366}(279,\cdot)\)
\(\chi_{2366}(293,\cdot)\)
\(\chi_{2366}(349,\cdot)\)
\(\chi_{2366}(405,\cdot)\)
\(\chi_{2366}(461,\cdot)\)
\(\chi_{2366}(475,\cdot)\)
\(\chi_{2366}(531,\cdot)\)
\(\chi_{2366}(643,\cdot)\)
\(\chi_{2366}(713,\cdot)\)
\(\chi_{2366}(769,\cdot)\)
\(\chi_{2366}(825,\cdot)\)
\(\chi_{2366}(839,\cdot)\)
\(\chi_{2366}(895,\cdot)\)
\(\chi_{2366}(951,\cdot)\)
\(\chi_{2366}(1007,\cdot)\)
\(\chi_{2366}(1021,\cdot)\)
\(\chi_{2366}(1077,\cdot)\)
\(\chi_{2366}(1133,\cdot)\)
\(\chi_{2366}(1189,\cdot)\)
\(\chi_{2366}(1203,\cdot)\)
\(\chi_{2366}(1259,\cdot)\)
\(\chi_{2366}(1315,\cdot)\)
\(\chi_{2366}(1385,\cdot)\)
\(\chi_{2366}(1497,\cdot)\)
\(\chi_{2366}(1553,\cdot)\)
...
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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));
|
| Fixed field: |
Number field defined by a degree 156 polynomial (not computed) |
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sage:chi.fixed_field()
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\((339,2199)\) → \((-1,e\left(\frac{29}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 2366 }(1007, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{23}{156}\right)\) | \(e\left(\frac{113}{156}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) |
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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)
