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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1220, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,15,11]))
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pari:[g,chi] = znchar(Mod(663,1220))
| Modulus: | \(1220\) | |
| Conductor: | \(1220\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(20\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1220}(23,\cdot)\)
\(\chi_{1220}(587,\cdot)\)
\(\chi_{1220}(643,\cdot)\)
\(\chi_{1220}(647,\cdot)\)
\(\chi_{1220}(663,\cdot)\)
\(\chi_{1220}(1167,\cdot)\)
\(\chi_{1220}(1183,\cdot)\)
\(\chi_{1220}(1187,\cdot)\)
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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 ]
\((611,977,1161)\) → \((-1,-i,e\left(\frac{11}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1220 }(663, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(-i\) | \(i\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(i\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) |
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sage:chi.jacobi_sum(n)
