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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10304, base_ring=CyclotomicField(176))
M = H._module
chi = DirichletCharacter(H, M([88,55,88,80]))
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gp:[g,chi] = znchar(Mod(3275, 10304))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10304.3275");
| Modulus: | \(10304\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(10304\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(176\) |
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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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\(\chi_{10304}(27,\cdot)\)
\(\chi_{10304}(307,\cdot)\)
\(\chi_{10304}(363,\cdot)\)
\(\chi_{10304}(531,\cdot)\)
\(\chi_{10304}(587,\cdot)\)
\(\chi_{10304}(699,\cdot)\)
\(\chi_{10304}(811,\cdot)\)
\(\chi_{10304}(867,\cdot)\)
\(\chi_{10304}(923,\cdot)\)
\(\chi_{10304}(979,\cdot)\)
\(\chi_{10304}(1315,\cdot)\)
\(\chi_{10304}(1595,\cdot)\)
\(\chi_{10304}(1651,\cdot)\)
\(\chi_{10304}(1819,\cdot)\)
\(\chi_{10304}(1875,\cdot)\)
\(\chi_{10304}(1987,\cdot)\)
\(\chi_{10304}(2099,\cdot)\)
\(\chi_{10304}(2155,\cdot)\)
\(\chi_{10304}(2211,\cdot)\)
\(\chi_{10304}(2267,\cdot)\)
\(\chi_{10304}(2603,\cdot)\)
\(\chi_{10304}(2883,\cdot)\)
\(\chi_{10304}(2939,\cdot)\)
\(\chi_{10304}(3107,\cdot)\)
\(\chi_{10304}(3163,\cdot)\)
\(\chi_{10304}(3275,\cdot)\)
\(\chi_{10304}(3387,\cdot)\)
\(\chi_{10304}(3443,\cdot)\)
\(\chi_{10304}(3499,\cdot)\)
\(\chi_{10304}(3555,\cdot)\)
...
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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: |
$\Q(\zeta_{176})$ |
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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 176 polynomial (not computed) |
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sage:chi.fixed_field()
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\((9983,645,1473,6721)\) → \((-1,e\left(\frac{5}{16}\right),-1,e\left(\frac{5}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
| \( \chi_{ 10304 }(3275, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{176}\right)\) | \(e\left(\frac{47}{176}\right)\) | \(e\left(\frac{37}{88}\right)\) | \(e\left(\frac{27}{176}\right)\) | \(e\left(\frac{97}{176}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{1}{176}\right)\) | \(e\left(\frac{47}{88}\right)\) | \(e\left(\frac{111}{176}\right)\) |
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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)
