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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5225, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([27,0,80]))
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gp:[g,chi] = znchar(Mod(408, 5225))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5225.408");
| Modulus: | \(5225\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(475\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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 \(\chi_{475}(408,\cdot)\) |
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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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\(\chi_{5225}(23,\cdot)\)
\(\chi_{5225}(177,\cdot)\)
\(\chi_{5225}(188,\cdot)\)
\(\chi_{5225}(397,\cdot)\)
\(\chi_{5225}(408,\cdot)\)
\(\chi_{5225}(617,\cdot)\)
\(\chi_{5225}(727,\cdot)\)
\(\chi_{5225}(738,\cdot)\)
\(\chi_{5225}(947,\cdot)\)
\(\chi_{5225}(1013,\cdot)\)
\(\chi_{5225}(1222,\cdot)\)
\(\chi_{5225}(1233,\cdot)\)
\(\chi_{5225}(1277,\cdot)\)
\(\chi_{5225}(1442,\cdot)\)
\(\chi_{5225}(1453,\cdot)\)
\(\chi_{5225}(1563,\cdot)\)
\(\chi_{5225}(1662,\cdot)\)
\(\chi_{5225}(1772,\cdot)\)
\(\chi_{5225}(1783,\cdot)\)
\(\chi_{5225}(1992,\cdot)\)
\(\chi_{5225}(2058,\cdot)\)
\(\chi_{5225}(2113,\cdot)\)
\(\chi_{5225}(2267,\cdot)\)
\(\chi_{5225}(2278,\cdot)\)
\(\chi_{5225}(2322,\cdot)\)
\(\chi_{5225}(2487,\cdot)\)
\(\chi_{5225}(2498,\cdot)\)
\(\chi_{5225}(2608,\cdot)\)
\(\chi_{5225}(2817,\cdot)\)
\(\chi_{5225}(2828,\cdot)\)
...
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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: |
$\Q(\zeta_{180})$ |
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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 180 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((2927,2851,4676)\) → \((e\left(\frac{3}{20}\right),1,e\left(\frac{4}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 5225 }(408, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{107}{180}\right)\) | \(e\left(\frac{149}{180}\right)\) | \(e\left(\frac{17}{90}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{59}{90}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{13}{180}\right)\) | \(e\left(\frac{1}{90}\right)\) |
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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)
