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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1040, base_ring=CyclotomicField(4))
M = H._module
chi = DirichletCharacter(H, M([0,1,2,2]))
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pari:[g,chi] = znchar(Mod(389,1040))
| Modulus: | \(1040\) | |
| Conductor: | \(1040\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(4\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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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\(\chi_{1040}(389,\cdot)\)
\(\chi_{1040}(909,\cdot)\)
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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 ]
\((911,261,417,561)\) → \((1,i,-1,-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 1040 }(389, a) \) |
\(1\) | \(1\) | \(i\) | \(-1\) | \(-1\) | \(-i\) | \(-1\) | \(i\) | \(-i\) | \(1\) | \(-i\) | \(-i\) |
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sage:chi.jacobi_sum(n)
