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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([17,22]))
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pari:[g,chi] = znchar(Mod(715,1156))
| 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
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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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\(\chi_{1156}(35,\cdot)\)
\(\chi_{1156}(103,\cdot)\)
\(\chi_{1156}(171,\cdot)\)
\(\chi_{1156}(239,\cdot)\)
\(\chi_{1156}(307,\cdot)\)
\(\chi_{1156}(375,\cdot)\)
\(\chi_{1156}(443,\cdot)\)
\(\chi_{1156}(511,\cdot)\)
\(\chi_{1156}(647,\cdot)\)
\(\chi_{1156}(715,\cdot)\)
\(\chi_{1156}(783,\cdot)\)
\(\chi_{1156}(851,\cdot)\)
\(\chi_{1156}(919,\cdot)\)
\(\chi_{1156}(987,\cdot)\)
\(\chi_{1156}(1055,\cdot)\)
\(\chi_{1156}(1123,\cdot)\)
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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 ]
\((579,581)\) → \((-1,e\left(\frac{11}{17}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 1156 }(715, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{27}{34}\right)\) |
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sage:chi.jacobi_sum(n)
