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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(91091, base_ring=CyclotomicField(546))
M = H._module
chi = DirichletCharacter(H, M([325,0,21]))
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gp:[g,chi] = znchar(Mod(1585, 91091))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("91091.1585");
| 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: | \(8281\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(546\) |
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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 \(\chi_{8281}(1585,\cdot)\) |
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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: | odd |
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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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\(\chi_{91091}(12,\cdot)\)
\(\chi_{91091}(584,\cdot)\)
\(\chi_{91091}(1585,\cdot)\)
\(\chi_{91091}(2014,\cdot)\)
\(\chi_{91091}(2586,\cdot)\)
\(\chi_{91091}(3015,\cdot)\)
\(\chi_{91091}(3587,\cdot)\)
\(\chi_{91091}(4016,\cdot)\)
\(\chi_{91091}(5589,\cdot)\)
\(\chi_{91091}(6018,\cdot)\)
\(\chi_{91091}(7019,\cdot)\)
\(\chi_{91091}(7591,\cdot)\)
\(\chi_{91091}(8020,\cdot)\)
\(\chi_{91091}(8592,\cdot)\)
\(\chi_{91091}(9021,\cdot)\)
\(\chi_{91091}(9593,\cdot)\)
\(\chi_{91091}(10022,\cdot)\)
\(\chi_{91091}(10594,\cdot)\)
\(\chi_{91091}(11023,\cdot)\)
\(\chi_{91091}(12596,\cdot)\)
\(\chi_{91091}(13025,\cdot)\)
\(\chi_{91091}(13597,\cdot)\)
\(\chi_{91091}(14598,\cdot)\)
\(\chi_{91091}(15027,\cdot)\)
\(\chi_{91091}(15599,\cdot)\)
\(\chi_{91091}(16028,\cdot)\)
\(\chi_{91091}(16600,\cdot)\)
\(\chi_{91091}(17029,\cdot)\)
\(\chi_{91091}(17601,\cdot)\)
\(\chi_{91091}(18030,\cdot)\)
...
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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: |
$\Q(\zeta_{273})$ |
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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 546 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((59489,41406,19944)\) → \((e\left(\frac{25}{42}\right),1,e\left(\frac{1}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(15\) |
| \( \chi_{ 91091 }(1585, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{281}{546}\right)\) | \(e\left(\frac{199}{546}\right)\) | \(e\left(\frac{8}{273}\right)\) | \(e\left(\frac{166}{273}\right)\) | \(e\left(\frac{80}{91}\right)\) | \(e\left(\frac{99}{182}\right)\) | \(e\left(\frac{199}{273}\right)\) | \(e\left(\frac{67}{546}\right)\) | \(e\left(\frac{215}{546}\right)\) | \(e\left(\frac{177}{182}\right)\) |
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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)
