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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4356, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([165,55,174]))
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gp:[g,chi] = znchar(Mod(839, 4356))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4356.839");
| Modulus: | \(4356\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4356\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(330\) |
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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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\(\chi_{4356}(47,\cdot)\)
\(\chi_{4356}(59,\cdot)\)
\(\chi_{4356}(119,\cdot)\)
\(\chi_{4356}(191,\cdot)\)
\(\chi_{4356}(203,\cdot)\)
\(\chi_{4356}(311,\cdot)\)
\(\chi_{4356}(335,\cdot)\)
\(\chi_{4356}(383,\cdot)\)
\(\chi_{4356}(443,\cdot)\)
\(\chi_{4356}(455,\cdot)\)
\(\chi_{4356}(515,\cdot)\)
\(\chi_{4356}(587,\cdot)\)
\(\chi_{4356}(599,\cdot)\)
\(\chi_{4356}(707,\cdot)\)
\(\chi_{4356}(731,\cdot)\)
\(\chi_{4356}(779,\cdot)\)
\(\chi_{4356}(839,\cdot)\)
\(\chi_{4356}(851,\cdot)\)
\(\chi_{4356}(911,\cdot)\)
\(\chi_{4356}(983,\cdot)\)
\(\chi_{4356}(1103,\cdot)\)
\(\chi_{4356}(1127,\cdot)\)
\(\chi_{4356}(1175,\cdot)\)
\(\chi_{4356}(1235,\cdot)\)
\(\chi_{4356}(1247,\cdot)\)
\(\chi_{4356}(1307,\cdot)\)
\(\chi_{4356}(1379,\cdot)\)
\(\chi_{4356}(1391,\cdot)\)
\(\chi_{4356}(1499,\cdot)\)
\(\chi_{4356}(1523,\cdot)\)
...
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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: |
$\Q(\zeta_{165})$ |
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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 330 polynomial (not computed) |
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sage:chi.fixed_field()
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\((2179,1937,1333)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{29}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4356 }(839, a) \) |
\(1\) | \(1\) | \(e\left(\frac{281}{330}\right)\) | \(e\left(\frac{283}{330}\right)\) | \(e\left(\frac{97}{165}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{116}{165}\right)\) | \(e\left(\frac{43}{330}\right)\) | \(e\left(\frac{59}{330}\right)\) | \(e\left(\frac{39}{55}\right)\) |
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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)
