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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(37950, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([0,165,198,150]))
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gp:[g,chi] = znchar(Mod(10093, 37950))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("37950.10093");
| Modulus: | \(37950\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1265\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(220\) |
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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 \(\chi_{1265}(1238,\cdot)\) |
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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);
|
\(\chi_{37950}(7,\cdot)\)
\(\chi_{37950}(343,\cdot)\)
\(\chi_{37950}(457,\cdot)\)
\(\chi_{37950}(1207,\cdot)\)
\(\chi_{37950}(1993,\cdot)\)
\(\chi_{37950}(2107,\cdot)\)
\(\chi_{37950}(2593,\cdot)\)
\(\chi_{37950}(2857,\cdot)\)
\(\chi_{37950}(3043,\cdot)\)
\(\chi_{37950}(3493,\cdot)\)
\(\chi_{37950}(3907,\cdot)\)
\(\chi_{37950}(4243,\cdot)\)
\(\chi_{37950}(5143,\cdot)\)
\(\chi_{37950}(5557,\cdot)\)
\(\chi_{37950}(5893,\cdot)\)
\(\chi_{37950}(6943,\cdot)\)
\(\chi_{37950}(7057,\cdot)\)
\(\chi_{37950}(7807,\cdot)\)
\(\chi_{37950}(8593,\cdot)\)
\(\chi_{37950}(9907,\cdot)\)
\(\chi_{37950}(10093,\cdot)\)
\(\chi_{37950}(10357,\cdot)\)
\(\chi_{37950}(10507,\cdot)\)
\(\chi_{37950}(10843,\cdot)\)
\(\chi_{37950}(11107,\cdot)\)
\(\chi_{37950}(11557,\cdot)\)
\(\chi_{37950}(12757,\cdot)\)
\(\chi_{37950}(12943,\cdot)\)
\(\chi_{37950}(13207,\cdot)\)
\(\chi_{37950}(13393,\cdot)\)
...
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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 };
\((25301,10627,27601,11551)\) → \((1,-i,e\left(\frac{9}{10}\right),e\left(\frac{15}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 37950 }(10093, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{220}\right)\) | \(e\left(\frac{153}{220}\right)\) | \(e\left(\frac{137}{220}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{191}{220}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{19}{20}\right)\) |
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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)
