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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([0,18,26,9]))
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pari:[g,chi] = znchar(Mod(1037,1080))
| 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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1080}(77,\cdot)\)
\(\chi_{1080}(173,\cdot)\)
\(\chi_{1080}(293,\cdot)\)
\(\chi_{1080}(317,\cdot)\)
\(\chi_{1080}(437,\cdot)\)
\(\chi_{1080}(533,\cdot)\)
\(\chi_{1080}(653,\cdot)\)
\(\chi_{1080}(677,\cdot)\)
\(\chi_{1080}(797,\cdot)\)
\(\chi_{1080}(893,\cdot)\)
\(\chi_{1080}(1013,\cdot)\)
\(\chi_{1080}(1037,\cdot)\)
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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 ]
\((271,541,1001,217)\) → \((1,-1,e\left(\frac{13}{18}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1080 }(1037, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{18}\right)\) |
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sage:chi.jacobi_sum(n)
