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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1984, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,3,8]))
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pari:[g,chi] = znchar(Mod(1979,1984))
| Modulus: | \(1984\) | |
| Conductor: | \(1984\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1984}(99,\cdot)\)
\(\chi_{1984}(243,\cdot)\)
\(\chi_{1984}(347,\cdot)\)
\(\chi_{1984}(491,\cdot)\)
\(\chi_{1984}(595,\cdot)\)
\(\chi_{1984}(739,\cdot)\)
\(\chi_{1984}(843,\cdot)\)
\(\chi_{1984}(987,\cdot)\)
\(\chi_{1984}(1091,\cdot)\)
\(\chi_{1984}(1235,\cdot)\)
\(\chi_{1984}(1339,\cdot)\)
\(\chi_{1984}(1483,\cdot)\)
\(\chi_{1984}(1587,\cdot)\)
\(\chi_{1984}(1731,\cdot)\)
\(\chi_{1984}(1835,\cdot)\)
\(\chi_{1984}(1979,\cdot)\)
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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 ]
\((63,1861,65)\) → \((-1,e\left(\frac{1}{16}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1984 }(1979, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(i\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) |
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sage:chi.jacobi_sum(n)
