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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8374, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([45,14]))
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gp:[g,chi] = znchar(Mod(2187, 8374))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8374.2187");
| Modulus: | \(8374\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4187\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(156\) |
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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 \(\chi_{4187}(2187,\cdot)\) |
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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: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{8374}(3,\cdot)\)
\(\chi_{8374}(243,\cdot)\)
\(\chi_{8374}(509,\cdot)\)
\(\chi_{8374}(511,\cdot)\)
\(\chi_{8374}(639,\cdot)\)
\(\chi_{8374}(951,\cdot)\)
\(\chi_{8374}(987,\cdot)\)
\(\chi_{8374}(1293,\cdot)\)
\(\chi_{8374}(1311,\cdot)\)
\(\chi_{8374}(1373,\cdot)\)
\(\chi_{8374}(1587,\cdot)\)
\(\chi_{8374}(1655,\cdot)\)
\(\chi_{8374}(1665,\cdot)\)
\(\chi_{8374}(2181,\cdot)\)
\(\chi_{8374}(2187,\cdot)\)
\(\chi_{8374}(2247,\cdot)\)
\(\chi_{8374}(2351,\cdot)\)
\(\chi_{8374}(2359,\cdot)\)
\(\chi_{8374}(2517,\cdot)\)
\(\chi_{8374}(2605,\cdot)\)
\(\chi_{8374}(2881,\cdot)\)
\(\chi_{8374}(2897,\cdot)\)
\(\chi_{8374}(3041,\cdot)\)
\(\chi_{8374}(3425,\cdot)\)
\(\chi_{8374}(4063,\cdot)\)
\(\chi_{8374}(4411,\cdot)\)
\(\chi_{8374}(4483,\cdot)\)
\(\chi_{8374}(4709,\cdot)\)
\(\chi_{8374}(5133,\cdot)\)
\(\chi_{8374}(5261,\cdot)\)
...
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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 };
\((8217,319)\) → \((e\left(\frac{15}{52}\right),e\left(\frac{7}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 8374 }(2187, a) \) |
\(1\) | \(1\) | \(e\left(\frac{155}{156}\right)\) | \(e\left(\frac{19}{156}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{85}{156}\right)\) | \(e\left(\frac{41}{52}\right)\) |
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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)
