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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1156, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,11]))
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pari:[g,chi] = znchar(Mod(1087,1156))
| Modulus: | \(1156\) | |
| Conductor: | \(1156\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(34\) |
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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
|
| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1156}(67,\cdot)\)
\(\chi_{1156}(135,\cdot)\)
\(\chi_{1156}(203,\cdot)\)
\(\chi_{1156}(271,\cdot)\)
\(\chi_{1156}(339,\cdot)\)
\(\chi_{1156}(407,\cdot)\)
\(\chi_{1156}(475,\cdot)\)
\(\chi_{1156}(543,\cdot)\)
\(\chi_{1156}(611,\cdot)\)
\(\chi_{1156}(679,\cdot)\)
\(\chi_{1156}(747,\cdot)\)
\(\chi_{1156}(815,\cdot)\)
\(\chi_{1156}(883,\cdot)\)
\(\chi_{1156}(951,\cdot)\)
\(\chi_{1156}(1019,\cdot)\)
\(\chi_{1156}(1087,\cdot)\)
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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 ]
\((579,581)\) → \((-1,e\left(\frac{11}{34}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 1156 }(1087, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) |
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sage:chi.jacobi_sum(n)
