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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([63,54,50]))
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gp:[g,chi] = znchar(Mod(3528, 5225))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5225.3528");
| 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: | \(5225\) |
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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: | 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: | 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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\(\chi_{5225}(13,\cdot)\)
\(\chi_{5225}(117,\cdot)\)
\(\chi_{5225}(127,\cdot)\)
\(\chi_{5225}(288,\cdot)\)
\(\chi_{5225}(402,\cdot)\)
\(\chi_{5225}(458,\cdot)\)
\(\chi_{5225}(523,\cdot)\)
\(\chi_{5225}(547,\cdot)\)
\(\chi_{5225}(667,\cdot)\)
\(\chi_{5225}(838,\cdot)\)
\(\chi_{5225}(952,\cdot)\)
\(\chi_{5225}(1097,\cdot)\)
\(\chi_{5225}(1212,\cdot)\)
\(\chi_{5225}(1283,\cdot)\)
\(\chi_{5225}(1492,\cdot)\)
\(\chi_{5225}(1647,\cdot)\)
\(\chi_{5225}(1663,\cdot)\)
\(\chi_{5225}(1762,\cdot)\)
\(\chi_{5225}(1777,\cdot)\)
\(\chi_{5225}(1922,\cdot)\)
\(\chi_{5225}(2312,\cdot)\)
\(\chi_{5225}(2428,\cdot)\)
\(\chi_{5225}(2472,\cdot)\)
\(\chi_{5225}(2587,\cdot)\)
\(\chi_{5225}(2978,\cdot)\)
\(\chi_{5225}(2998,\cdot)\)
\(\chi_{5225}(3137,\cdot)\)
\(\chi_{5225}(3297,\cdot)\)
\(\chi_{5225}(3528,\cdot)\)
\(\chi_{5225}(3548,\cdot)\)
...
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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 };
\((2927,2851,4676)\) → \((e\left(\frac{7}{20}\right),e\left(\frac{3}{10}\right),e\left(\frac{5}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 5225 }(3528, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{167}{180}\right)\) | \(e\left(\frac{83}{180}\right)\) | \(e\left(\frac{77}{90}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{83}{90}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{61}{180}\right)\) | \(e\left(\frac{4}{9}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
