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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5160, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([7,7,7,7,6]))
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pari:[g,chi] = znchar(Mod(2099,5160))
| Modulus: | \(5160\) | |
| Conductor: | \(5160\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(14\) |
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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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\(\chi_{5160}(59,\cdot)\)
\(\chi_{5160}(299,\cdot)\)
\(\chi_{5160}(1139,\cdot)\)
\(\chi_{5160}(2099,\cdot)\)
\(\chi_{5160}(3659,\cdot)\)
\(\chi_{5160}(4139,\cdot)\)
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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 ]
\((3871,2581,1721,3097,4561)\) → \((-1,-1,-1,-1,e\left(\frac{3}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5160 }(2099, a) \) |
\(1\) | \(1\) | \(1\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(1\) | \(e\left(\frac{1}{14}\right)\) |
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sage:chi.jacobi_sum(n)
