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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3960, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,30,50,45,18]))
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pari:[g,chi] = znchar(Mod(2813,3960))
| Modulus: | \(3960\) | |
| Conductor: | \(3960\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(60\) |
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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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\(\chi_{3960}(173,\cdot)\)
\(\chi_{3960}(293,\cdot)\)
\(\chi_{3960}(437,\cdot)\)
\(\chi_{3960}(677,\cdot)\)
\(\chi_{3960}(893,\cdot)\)
\(\chi_{3960}(1157,\cdot)\)
\(\chi_{3960}(1613,\cdot)\)
\(\chi_{3960}(1733,\cdot)\)
\(\chi_{3960}(1757,\cdot)\)
\(\chi_{3960}(1877,\cdot)\)
\(\chi_{3960}(2477,\cdot)\)
\(\chi_{3960}(2813,\cdot)\)
\(\chi_{3960}(3053,\cdot)\)
\(\chi_{3960}(3197,\cdot)\)
\(\chi_{3960}(3317,\cdot)\)
\(\chi_{3960}(3533,\cdot)\)
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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 ]
\((991,1981,3521,2377,2521)\) → \((1,-1,e\left(\frac{5}{6}\right),-i,e\left(\frac{3}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 3960 }(2813, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{7}{12}\right)\) |
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sage:chi.jacobi_sum(n)
