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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3600, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,15,50,12]))
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pari:[g,chi] = znchar(Mod(491,3600))
| Modulus: | \(3600\) | |
| Conductor: | \(3600\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| 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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{3600}(11,\cdot)\)
\(\chi_{3600}(131,\cdot)\)
\(\chi_{3600}(371,\cdot)\)
\(\chi_{3600}(491,\cdot)\)
\(\chi_{3600}(731,\cdot)\)
\(\chi_{3600}(1091,\cdot)\)
\(\chi_{3600}(1211,\cdot)\)
\(\chi_{3600}(1571,\cdot)\)
\(\chi_{3600}(1811,\cdot)\)
\(\chi_{3600}(1931,\cdot)\)
\(\chi_{3600}(2171,\cdot)\)
\(\chi_{3600}(2291,\cdot)\)
\(\chi_{3600}(2531,\cdot)\)
\(\chi_{3600}(2891,\cdot)\)
\(\chi_{3600}(3011,\cdot)\)
\(\chi_{3600}(3371,\cdot)\)
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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 ]
\((3151,901,2801,577)\) → \((-1,i,e\left(\frac{5}{6}\right),e\left(\frac{1}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3600 }(491, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) |
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sage:chi.jacobi_sum(n)
