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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7616, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,21,8,36]))
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pari:[g,chi] = znchar(Mod(7059,7616))
| Modulus: | \(7616\) | |
| Conductor: | \(7616\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(48\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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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\(\chi_{7616}(523,\cdot)\)
\(\chi_{7616}(803,\cdot)\)
\(\chi_{7616}(1067,\cdot)\)
\(\chi_{7616}(1347,\cdot)\)
\(\chi_{7616}(2427,\cdot)\)
\(\chi_{7616}(2707,\cdot)\)
\(\chi_{7616}(2971,\cdot)\)
\(\chi_{7616}(3251,\cdot)\)
\(\chi_{7616}(4331,\cdot)\)
\(\chi_{7616}(4611,\cdot)\)
\(\chi_{7616}(4875,\cdot)\)
\(\chi_{7616}(5155,\cdot)\)
\(\chi_{7616}(6235,\cdot)\)
\(\chi_{7616}(6515,\cdot)\)
\(\chi_{7616}(6779,\cdot)\)
\(\chi_{7616}(7059,\cdot)\)
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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 ]
\((5951,3333,3265,2689)\) → \((-1,e\left(\frac{7}{16}\right),e\left(\frac{1}{6}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 7616 }(7059, a) \) |
\(1\) | \(1\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(-i\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{3}{16}\right)\) |
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sage:chi.jacobi_sum(n)
