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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(72000, base_ring=CyclotomicField(400))
M = H._module
chi = DirichletCharacter(H, M([200,175,0,352]))
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gp:[g,chi] = znchar(Mod(6931, 72000))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("72000.6931");
| Modulus: | \(72000\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8000\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(400\) |
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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 \(\chi_{8000}(6931,\cdot)\) |
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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);
|
\(\chi_{72000}(91,\cdot)\)
\(\chi_{72000}(811,\cdot)\)
\(\chi_{72000}(1171,\cdot)\)
\(\chi_{72000}(1531,\cdot)\)
\(\chi_{72000}(1891,\cdot)\)
\(\chi_{72000}(2611,\cdot)\)
\(\chi_{72000}(2971,\cdot)\)
\(\chi_{72000}(3331,\cdot)\)
\(\chi_{72000}(3691,\cdot)\)
\(\chi_{72000}(4411,\cdot)\)
\(\chi_{72000}(4771,\cdot)\)
\(\chi_{72000}(5131,\cdot)\)
\(\chi_{72000}(5491,\cdot)\)
\(\chi_{72000}(6211,\cdot)\)
\(\chi_{72000}(6571,\cdot)\)
\(\chi_{72000}(6931,\cdot)\)
\(\chi_{72000}(7291,\cdot)\)
\(\chi_{72000}(8011,\cdot)\)
\(\chi_{72000}(8371,\cdot)\)
\(\chi_{72000}(8731,\cdot)\)
\(\chi_{72000}(9091,\cdot)\)
\(\chi_{72000}(9811,\cdot)\)
\(\chi_{72000}(10171,\cdot)\)
\(\chi_{72000}(10531,\cdot)\)
\(\chi_{72000}(10891,\cdot)\)
\(\chi_{72000}(11611,\cdot)\)
\(\chi_{72000}(11971,\cdot)\)
\(\chi_{72000}(12331,\cdot)\)
\(\chi_{72000}(12691,\cdot)\)
\(\chi_{72000}(13411,\cdot)\)
...
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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 };
\((42751,58501,64001,29377)\) → \((-1,e\left(\frac{7}{16}\right),1,e\left(\frac{22}{25}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 72000 }(6931, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{227}{400}\right)\) | \(e\left(\frac{353}{400}\right)\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{161}{400}\right)\) | \(e\left(\frac{181}{200}\right)\) | \(e\left(\frac{149}{400}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{183}{400}\right)\) | \(e\left(\frac{169}{200}\right)\) |
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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)
