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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(50160, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([0,90,90,135,144,140]))
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gp:[g,chi] = znchar(Mod(6713, 50160))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("50160.6713");
| Modulus: | \(50160\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(25080\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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 \(\chi_{25080}(19253,\cdot)\) |
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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);
|
\(\chi_{50160}(137,\cdot)\)
\(\chi_{50160}(377,\cdot)\)
\(\chi_{50160}(2297,\cdot)\)
\(\chi_{50160}(3353,\cdot)\)
\(\chi_{50160}(4937,\cdot)\)
\(\chi_{50160}(6713,\cdot)\)
\(\chi_{50160}(6857,\cdot)\)
\(\chi_{50160}(7913,\cdot)\)
\(\chi_{50160}(9353,\cdot)\)
\(\chi_{50160}(9497,\cdot)\)
\(\chi_{50160}(11273,\cdot)\)
\(\chi_{50160}(11993,\cdot)\)
\(\chi_{50160}(13913,\cdot)\)
\(\chi_{50160}(15833,\cdot)\)
\(\chi_{50160}(15977,\cdot)\)
\(\chi_{50160}(16553,\cdot)\)
\(\chi_{50160}(17033,\cdot)\)
\(\chi_{50160}(18473,\cdot)\)
\(\chi_{50160}(18617,\cdot)\)
\(\chi_{50160}(18857,\cdot)\)
\(\chi_{50160}(21113,\cdot)\)
\(\chi_{50160}(23417,\cdot)\)
\(\chi_{50160}(24953,\cdot)\)
\(\chi_{50160}(26777,\cdot)\)
\(\chi_{50160}(27593,\cdot)\)
\(\chi_{50160}(27833,\cdot)\)
\(\chi_{50160}(27977,\cdot)\)
\(\chi_{50160}(29417,\cdot)\)
\(\chi_{50160}(30233,\cdot)\)
\(\chi_{50160}(30473,\cdot)\)
...
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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 };
\((6271,37621,16721,30097,9121,47521)\) → \((1,-1,-1,-i,e\left(\frac{4}{5}\right),e\left(\frac{7}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 50160 }(6713, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{79}{180}\right)\) | \(e\left(\frac{41}{180}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{90}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{157}{180}\right)\) |
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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)
