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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(58081, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([21]))
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gp:[g,chi] = znchar(Mod(32514, 58081))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("58081.32514");
| Modulus: | \(58081\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(241\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(80\) |
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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 \(\chi_{241}(220,\cdot)\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{58081}(583,\cdot)\)
\(\chi_{58081}(3477,\cdot)\)
\(\chi_{58081}(7976,\cdot)\)
\(\chi_{58081}(9175,\cdot)\)
\(\chi_{58081}(12198,\cdot)\)
\(\chi_{58081}(17085,\cdot)\)
\(\chi_{58081}(18283,\cdot)\)
\(\chi_{58081}(19124,\cdot)\)
\(\chi_{58081}(19626,\cdot)\)
\(\chi_{58081}(19946,\cdot)\)
\(\chi_{58081}(20995,\cdot)\)
\(\chi_{58081}(21733,\cdot)\)
\(\chi_{58081}(24684,\cdot)\)
\(\chi_{58081}(25567,\cdot)\)
\(\chi_{58081}(28080,\cdot)\)
\(\chi_{58081}(28124,\cdot)\)
\(\chi_{58081}(29957,\cdot)\)
\(\chi_{58081}(30001,\cdot)\)
\(\chi_{58081}(32514,\cdot)\)
\(\chi_{58081}(33397,\cdot)\)
\(\chi_{58081}(36348,\cdot)\)
\(\chi_{58081}(37086,\cdot)\)
\(\chi_{58081}(38135,\cdot)\)
\(\chi_{58081}(38455,\cdot)\)
\(\chi_{58081}(38957,\cdot)\)
\(\chi_{58081}(39798,\cdot)\)
\(\chi_{58081}(40996,\cdot)\)
\(\chi_{58081}(45883,\cdot)\)
\(\chi_{58081}(48906,\cdot)\)
\(\chi_{58081}(50105,\cdot)\)
...
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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 };
\(7\) → \(e\left(\frac{21}{80}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 58081 }(32514, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(-i\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{16}\right)\) |
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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)
