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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([24,3,8,33]))
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pari:[g,chi] = znchar(Mod(6331,7616))
| Modulus: | \(7616\) | |
| Conductor: | \(7616\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{7616}(3,\cdot)\)
\(\chi_{7616}(243,\cdot)\)
\(\chi_{7616}(1979,\cdot)\)
\(\chi_{7616}(2187,\cdot)\)
\(\chi_{7616}(2203,\cdot)\)
\(\chi_{7616}(2539,\cdot)\)
\(\chi_{7616}(2579,\cdot)\)
\(\chi_{7616}(3267,\cdot)\)
\(\chi_{7616}(3363,\cdot)\)
\(\chi_{7616}(4595,\cdot)\)
\(\chi_{7616}(5451,\cdot)\)
\(\chi_{7616}(5843,\cdot)\)
\(\chi_{7616}(6331,\cdot)\)
\(\chi_{7616}(6555,\cdot)\)
\(\chi_{7616}(6627,\cdot)\)
\(\chi_{7616}(6891,\cdot)\)
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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 ]
\((5951,3333,3265,2689)\) → \((-1,e\left(\frac{1}{16}\right),e\left(\frac{1}{6}\right),e\left(\frac{11}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 7616 }(6331, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{8}\right)\) |
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sage:chi.jacobi_sum(n)
