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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(91091, base_ring=CyclotomicField(2730))
M = H._module
chi = DirichletCharacter(H, M([1040,2184,315]))
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gp:[g,chi] = znchar(Mod(25, 91091))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("91091.25");
| Modulus: | \(91091\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(91091\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(2730\) |
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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);
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| 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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\(\chi_{91091}(25,\cdot)\)
\(\chi_{91091}(207,\cdot)\)
\(\chi_{91091}(389,\cdot)\)
\(\chi_{91091}(597,\cdot)\)
\(\chi_{91091}(779,\cdot)\)
\(\chi_{91091}(1026,\cdot)\)
\(\chi_{91091}(1208,\cdot)\)
\(\chi_{91091}(1325,\cdot)\)
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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: |
$\Q(\zeta_{1365})$ |
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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 2730 polynomial (not computed) |
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sage:chi.fixed_field()
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\((59489,41406,19944)\) → \((e\left(\frac{8}{21}\right),e\left(\frac{4}{5}\right),e\left(\frac{3}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(15\) |
| \( \chi_{ 91091 }(25, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2239}{2730}\right)\) | \(e\left(\frac{121}{1365}\right)\) | \(e\left(\frac{874}{1365}\right)\) | \(e\left(\frac{781}{2730}\right)\) | \(e\left(\frac{827}{910}\right)\) | \(e\left(\frac{419}{910}\right)\) | \(e\left(\frac{242}{1365}\right)\) | \(e\left(\frac{29}{273}\right)\) | \(e\left(\frac{199}{273}\right)\) | \(e\left(\frac{341}{910}\right)\) |
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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)
