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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4788, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([9,6,9,7]))
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pari:[g,chi] = znchar(Mod(3415,4788))
| Modulus: | \(4788\) | |
| Conductor: | \(4788\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(18\) |
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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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\(\chi_{4788}(895,\cdot)\)
\(\chi_{4788}(979,\cdot)\)
\(\chi_{4788}(1231,\cdot)\)
\(\chi_{4788}(1903,\cdot)\)
\(\chi_{4788}(3415,\cdot)\)
\(\chi_{4788}(4003,\cdot)\)
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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 ]
\((2395,533,4105,1009)\) → \((-1,e\left(\frac{1}{3}\right),-1,e\left(\frac{7}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4788 }(3415, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{18}\right)\) | \(-1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{2}{9}\right)\) |
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sage:chi.jacobi_sum(n)
