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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([18,9,20,21]))
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pari:[g,chi] = znchar(Mod(1051,5616))
| 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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\(\chi_{5616}(643,\cdot)\)
\(\chi_{5616}(691,\cdot)\)
\(\chi_{5616}(1051,\cdot)\)
\(\chi_{5616}(1723,\cdot)\)
\(\chi_{5616}(2515,\cdot)\)
\(\chi_{5616}(2563,\cdot)\)
\(\chi_{5616}(2923,\cdot)\)
\(\chi_{5616}(3595,\cdot)\)
\(\chi_{5616}(4387,\cdot)\)
\(\chi_{5616}(4435,\cdot)\)
\(\chi_{5616}(4795,\cdot)\)
\(\chi_{5616}(5467,\cdot)\)
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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 ]
\((703,4213,2081,3889)\) → \((-1,i,e\left(\frac{5}{9}\right),e\left(\frac{7}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 5616 }(1051, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) |
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sage:chi.jacobi_sum(n)
