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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14160, base_ring=CyclotomicField(116))
M = H._module
chi = DirichletCharacter(H, M([0,87,58,0,32]))
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gp:[g,chi] = znchar(Mod(7421, 14160))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("14160.7421");
| Modulus: | \(14160\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2832\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(116\) |
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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 \(\chi_{2832}(1757,\cdot)\) |
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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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\(\chi_{14160}(341,\cdot)\)
\(\chi_{14160}(461,\cdot)\)
\(\chi_{14160}(1301,\cdot)\)
\(\chi_{14160}(1421,\cdot)\)
\(\chi_{14160}(1541,\cdot)\)
\(\chi_{14160}(1661,\cdot)\)
\(\chi_{14160}(2021,\cdot)\)
\(\chi_{14160}(2141,\cdot)\)
\(\chi_{14160}(2261,\cdot)\)
\(\chi_{14160}(2381,\cdot)\)
\(\chi_{14160}(2621,\cdot)\)
\(\chi_{14160}(2741,\cdot)\)
\(\chi_{14160}(2861,\cdot)\)
\(\chi_{14160}(3221,\cdot)\)
\(\chi_{14160}(3581,\cdot)\)
\(\chi_{14160}(3821,\cdot)\)
\(\chi_{14160}(4061,\cdot)\)
\(\chi_{14160}(4181,\cdot)\)
\(\chi_{14160}(4301,\cdot)\)
\(\chi_{14160}(4541,\cdot)\)
\(\chi_{14160}(4901,\cdot)\)
\(\chi_{14160}(5381,\cdot)\)
\(\chi_{14160}(5621,\cdot)\)
\(\chi_{14160}(5861,\cdot)\)
\(\chi_{14160}(5981,\cdot)\)
\(\chi_{14160}(6221,\cdot)\)
\(\chi_{14160}(6341,\cdot)\)
\(\chi_{14160}(6821,\cdot)\)
\(\chi_{14160}(7421,\cdot)\)
\(\chi_{14160}(7541,\cdot)\)
...
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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 };
\((5311,3541,4721,8497,3601)\) → \((1,-i,-1,1,e\left(\frac{8}{29}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 14160 }(7421, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{27}{58}\right)\) | \(e\left(\frac{17}{116}\right)\) | \(e\left(\frac{77}{116}\right)\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{85}{116}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{55}{116}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{107}{116}\right)\) | \(e\left(\frac{25}{29}\right)\) |
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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)
