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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2496, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,27,24,8]))
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pari:[g,chi] = znchar(Mod(1499,2496))
| Modulus: | \(2496\) | |
| Conductor: | \(2496\) |
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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)
|
| 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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\(\chi_{2496}(179,\cdot)\)
\(\chi_{2496}(251,\cdot)\)
\(\chi_{2496}(491,\cdot)\)
\(\chi_{2496}(563,\cdot)\)
\(\chi_{2496}(803,\cdot)\)
\(\chi_{2496}(875,\cdot)\)
\(\chi_{2496}(1115,\cdot)\)
\(\chi_{2496}(1187,\cdot)\)
\(\chi_{2496}(1427,\cdot)\)
\(\chi_{2496}(1499,\cdot)\)
\(\chi_{2496}(1739,\cdot)\)
\(\chi_{2496}(1811,\cdot)\)
\(\chi_{2496}(2051,\cdot)\)
\(\chi_{2496}(2123,\cdot)\)
\(\chi_{2496}(2363,\cdot)\)
\(\chi_{2496}(2435,\cdot)\)
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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 ]
\((703,1093,833,769)\) → \((-1,e\left(\frac{9}{16}\right),-1,e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2496 }(1499, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(-1\) | \(e\left(\frac{25}{48}\right)\) |
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sage:chi.jacobi_sum(n)
