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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5616, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,2,3]))
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pari:[g,chi] = znchar(Mod(4565,5616))
| Modulus: | \(5616\) | |
| Conductor: | \(5616\) |
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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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\(\chi_{5616}(149,\cdot)\)
\(\chi_{5616}(821,\cdot)\)
\(\chi_{5616}(1181,\cdot)\)
\(\chi_{5616}(1229,\cdot)\)
\(\chi_{5616}(2021,\cdot)\)
\(\chi_{5616}(2693,\cdot)\)
\(\chi_{5616}(3053,\cdot)\)
\(\chi_{5616}(3101,\cdot)\)
\(\chi_{5616}(3893,\cdot)\)
\(\chi_{5616}(4565,\cdot)\)
\(\chi_{5616}(4925,\cdot)\)
\(\chi_{5616}(4973,\cdot)\)
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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 ]
\((703,4213,2081,3889)\) → \((1,i,e\left(\frac{1}{18}\right),e\left(\frac{1}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 5616 }(4565, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) |
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sage:chi.jacobi_sum(n)
