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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([18,9,16,15]))
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pari:[g,chi] = znchar(Mod(1579,5616))
| 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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\(\chi_{5616}(115,\cdot)\)
\(\chi_{5616}(787,\cdot)\)
\(\chi_{5616}(1579,\cdot)\)
\(\chi_{5616}(1627,\cdot)\)
\(\chi_{5616}(1987,\cdot)\)
\(\chi_{5616}(2659,\cdot)\)
\(\chi_{5616}(3451,\cdot)\)
\(\chi_{5616}(3499,\cdot)\)
\(\chi_{5616}(3859,\cdot)\)
\(\chi_{5616}(4531,\cdot)\)
\(\chi_{5616}(5323,\cdot)\)
\(\chi_{5616}(5371,\cdot)\)
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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 ]
\((703,4213,2081,3889)\) → \((-1,i,e\left(\frac{4}{9}\right),e\left(\frac{5}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 5616 }(1579, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) |
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sage:chi.jacobi_sum(n)
