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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1368, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([0,9,6,1]))
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pari:[g,chi] = znchar(Mod(1237,1368))
| Modulus: | \(1368\) | |
| Conductor: | \(1368\) |
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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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\(\chi_{1368}(205,\cdot)\)
\(\chi_{1368}(637,\cdot)\)
\(\chi_{1368}(877,\cdot)\)
\(\chi_{1368}(925,\cdot)\)
\(\chi_{1368}(1021,\cdot)\)
\(\chi_{1368}(1237,\cdot)\)
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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 ]
\((343,685,1217,1009)\) → \((1,-1,e\left(\frac{1}{3}\right),e\left(\frac{1}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1368 }(1237, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(-1\) | \(e\left(\frac{13}{18}\right)\) |
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sage:chi.jacobi_sum(n)
