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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1080, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,18,20,27]))
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pari:[g,chi] = znchar(Mod(133,1080))
| Modulus: | \(1080\) | |
| Conductor: | \(1080\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(36\) |
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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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\(\chi_{1080}(13,\cdot)\)
\(\chi_{1080}(133,\cdot)\)
\(\chi_{1080}(157,\cdot)\)
\(\chi_{1080}(277,\cdot)\)
\(\chi_{1080}(373,\cdot)\)
\(\chi_{1080}(493,\cdot)\)
\(\chi_{1080}(517,\cdot)\)
\(\chi_{1080}(637,\cdot)\)
\(\chi_{1080}(733,\cdot)\)
\(\chi_{1080}(853,\cdot)\)
\(\chi_{1080}(877,\cdot)\)
\(\chi_{1080}(997,\cdot)\)
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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 ]
\((271,541,1001,217)\) → \((1,-1,e\left(\frac{5}{9}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1080 }(133, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{4}{9}\right)\) |
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sage:chi.jacobi_sum(n)
