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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1620, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,53,27]))
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pari:[g,chi] = znchar(Mod(1499,1620))
| Modulus: | \(1620\) | |
| Conductor: | \(1620\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(54\) |
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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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\(\chi_{1620}(59,\cdot)\)
\(\chi_{1620}(119,\cdot)\)
\(\chi_{1620}(239,\cdot)\)
\(\chi_{1620}(299,\cdot)\)
\(\chi_{1620}(419,\cdot)\)
\(\chi_{1620}(479,\cdot)\)
\(\chi_{1620}(599,\cdot)\)
\(\chi_{1620}(659,\cdot)\)
\(\chi_{1620}(779,\cdot)\)
\(\chi_{1620}(839,\cdot)\)
\(\chi_{1620}(959,\cdot)\)
\(\chi_{1620}(1019,\cdot)\)
\(\chi_{1620}(1139,\cdot)\)
\(\chi_{1620}(1199,\cdot)\)
\(\chi_{1620}(1319,\cdot)\)
\(\chi_{1620}(1379,\cdot)\)
\(\chi_{1620}(1499,\cdot)\)
\(\chi_{1620}(1559,\cdot)\)
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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 ]
\((811,1541,1297)\) → \((-1,e\left(\frac{53}{54}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1620 }(1499, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{54}\right)\) |
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sage:chi.jacobi_sum(n)
