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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2368, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([72,99,140]))
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gp:[g,chi] = znchar(Mod(611, 2368))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2368.611");
| Modulus: | \(2368\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2368\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(144\) |
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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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\(\chi_{2368}(19,\cdot)\)
\(\chi_{2368}(35,\cdot)\)
\(\chi_{2368}(187,\cdot)\)
\(\chi_{2368}(203,\cdot)\)
\(\chi_{2368}(227,\cdot)\)
\(\chi_{2368}(283,\cdot)\)
\(\chi_{2368}(355,\cdot)\)
\(\chi_{2368}(387,\cdot)\)
\(\chi_{2368}(427,\cdot)\)
\(\chi_{2368}(459,\cdot)\)
\(\chi_{2368}(531,\cdot)\)
\(\chi_{2368}(587,\cdot)\)
\(\chi_{2368}(611,\cdot)\)
\(\chi_{2368}(627,\cdot)\)
\(\chi_{2368}(779,\cdot)\)
\(\chi_{2368}(795,\cdot)\)
\(\chi_{2368}(819,\cdot)\)
\(\chi_{2368}(875,\cdot)\)
\(\chi_{2368}(947,\cdot)\)
\(\chi_{2368}(979,\cdot)\)
\(\chi_{2368}(1019,\cdot)\)
\(\chi_{2368}(1051,\cdot)\)
\(\chi_{2368}(1123,\cdot)\)
\(\chi_{2368}(1179,\cdot)\)
\(\chi_{2368}(1203,\cdot)\)
\(\chi_{2368}(1219,\cdot)\)
\(\chi_{2368}(1371,\cdot)\)
\(\chi_{2368}(1387,\cdot)\)
\(\chi_{2368}(1411,\cdot)\)
\(\chi_{2368}(1467,\cdot)\)
...
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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: |
$\Q(\zeta_{144})$ |
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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 144 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((1407,1925,705)\) → \((-1,e\left(\frac{11}{16}\right),e\left(\frac{35}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2368 }(611, a) \) |
\(1\) | \(1\) | \(e\left(\frac{121}{144}\right)\) | \(e\left(\frac{7}{144}\right)\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{1}{144}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{49}{144}\right)\) | \(e\left(\frac{47}{144}\right)\) |
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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)
