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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4620, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,30,15,50,48]))
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pari:[g,chi] = znchar(Mod(47,4620))
| Modulus: | \(4620\) | |
| Conductor: | \(4620\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(60\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
|
| Real: | no |
| Primitive: | yes |
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sage:chi.is_primitive()
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pari:#znconreyconductor(g,chi)==1
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| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{4620}(47,\cdot)\)
\(\chi_{4620}(383,\cdot)\)
\(\chi_{4620}(467,\cdot)\)
\(\chi_{4620}(647,\cdot)\)
\(\chi_{4620}(983,\cdot)\)
\(\chi_{4620}(1307,\cdot)\)
\(\chi_{4620}(1643,\cdot)\)
\(\chi_{4620}(1907,\cdot)\)
\(\chi_{4620}(2567,\cdot)\)
\(\chi_{4620}(3083,\cdot)\)
\(\chi_{4620}(3503,\cdot)\)
\(\chi_{4620}(3743,\cdot)\)
\(\chi_{4620}(4007,\cdot)\)
\(\chi_{4620}(4163,\cdot)\)
\(\chi_{4620}(4343,\cdot)\)
\(\chi_{4620}(4427,\cdot)\)
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((2311,1541,3697,661,2521)\) → \((-1,-1,i,e\left(\frac{5}{6}\right),e\left(\frac{4}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 4620 }(47, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(i\) | \(e\left(\frac{49}{60}\right)\) |
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sage:chi.jacobi_sum(n)
