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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(78585, base_ring=CyclotomicField(780))
M = H._module
chi = DirichletCharacter(H, M([390,585,515,312]))
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gp:[g,chi] = znchar(Mod(5588, 78585))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("78585.5588");
| Modulus: | \(78585\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(78585\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(780\) |
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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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\(\chi_{78585}(2,\cdot)\)
\(\chi_{78585}(128,\cdot)\)
\(\chi_{78585}(977,\cdot)\)
\(\chi_{78585}(1397,\cdot)\)
\(\chi_{78585}(2048,\cdot)\)
\(\chi_{78585}(2147,\cdot)\)
\(\chi_{78585}(2372,\cdot)\)
\(\chi_{78585}(2732,\cdot)\)
\(\chi_{78585}(3443,\cdot)\)
\(\chi_{78585}(3542,\cdot)\)
\(\chi_{78585}(4127,\cdot)\)
\(\chi_{78585}(4193,\cdot)\)
\(\chi_{78585}(4418,\cdot)\)
\(\chi_{78585}(4778,\cdot)\)
\(\chi_{78585}(5588,\cdot)\)
\(\chi_{78585}(6047,\cdot)\)
\(\chi_{78585}(7022,\cdot)\)
\(\chi_{78585}(7442,\cdot)\)
\(\chi_{78585}(8417,\cdot)\)
\(\chi_{78585}(8777,\cdot)\)
\(\chi_{78585}(9068,\cdot)\)
\(\chi_{78585}(9488,\cdot)\)
\(\chi_{78585}(9587,\cdot)\)
\(\chi_{78585}(10172,\cdot)\)
\(\chi_{78585}(10238,\cdot)\)
\(\chi_{78585}(10463,\cdot)\)
\(\chi_{78585}(10823,\cdot)\)
\(\chi_{78585}(11633,\cdot)\)
\(\chi_{78585}(12092,\cdot)\)
\(\chi_{78585}(12218,\cdot)\)
...
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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 };
\((52391,47152,1861,32956)\) → \((-1,-i,e\left(\frac{103}{156}\right),e\left(\frac{2}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 78585 }(5588, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{199}{390}\right)\) | \(e\left(\frac{4}{195}\right)\) | \(e\left(\frac{233}{390}\right)\) | \(e\left(\frac{69}{130}\right)\) | \(e\left(\frac{551}{780}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{8}{195}\right)\) | \(e\left(\frac{349}{780}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) |
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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)
