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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5408, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([0,39,118]))
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gp:[g,chi] = znchar(Mod(985, 5408))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5408.985");
| Modulus: | \(5408\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2704\) |
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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 \(\chi_{2704}(309,\cdot)\) |
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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: | no |
| 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);
|
\(\chi_{5408}(121,\cdot)\)
\(\chi_{5408}(153,\cdot)\)
\(\chi_{5408}(329,\cdot)\)
\(\chi_{5408}(537,\cdot)\)
\(\chi_{5408}(569,\cdot)\)
\(\chi_{5408}(745,\cdot)\)
\(\chi_{5408}(777,\cdot)\)
\(\chi_{5408}(953,\cdot)\)
\(\chi_{5408}(985,\cdot)\)
\(\chi_{5408}(1193,\cdot)\)
\(\chi_{5408}(1369,\cdot)\)
\(\chi_{5408}(1401,\cdot)\)
\(\chi_{5408}(1577,\cdot)\)
\(\chi_{5408}(1609,\cdot)\)
\(\chi_{5408}(1785,\cdot)\)
\(\chi_{5408}(1817,\cdot)\)
\(\chi_{5408}(1993,\cdot)\)
\(\chi_{5408}(2025,\cdot)\)
\(\chi_{5408}(2201,\cdot)\)
\(\chi_{5408}(2233,\cdot)\)
\(\chi_{5408}(2409,\cdot)\)
\(\chi_{5408}(2441,\cdot)\)
\(\chi_{5408}(2617,\cdot)\)
\(\chi_{5408}(2649,\cdot)\)
\(\chi_{5408}(2825,\cdot)\)
\(\chi_{5408}(2857,\cdot)\)
\(\chi_{5408}(3033,\cdot)\)
\(\chi_{5408}(3241,\cdot)\)
\(\chi_{5408}(3273,\cdot)\)
\(\chi_{5408}(3449,\cdot)\)
...
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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 };
\((2367,677,1185)\) → \((1,i,e\left(\frac{59}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 5408 }(985, a) \) |
\(1\) | \(1\) | \(e\left(\frac{85}{156}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{25}{156}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{5}{6}\right)\) |
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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)
