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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([27,20,27]))
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pari:[g,chi] = znchar(Mod(679,1620))
| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1620}(79,\cdot)\)
\(\chi_{1620}(139,\cdot)\)
\(\chi_{1620}(259,\cdot)\)
\(\chi_{1620}(319,\cdot)\)
\(\chi_{1620}(439,\cdot)\)
\(\chi_{1620}(499,\cdot)\)
\(\chi_{1620}(619,\cdot)\)
\(\chi_{1620}(679,\cdot)\)
\(\chi_{1620}(799,\cdot)\)
\(\chi_{1620}(859,\cdot)\)
\(\chi_{1620}(979,\cdot)\)
\(\chi_{1620}(1039,\cdot)\)
\(\chi_{1620}(1159,\cdot)\)
\(\chi_{1620}(1219,\cdot)\)
\(\chi_{1620}(1339,\cdot)\)
\(\chi_{1620}(1399,\cdot)\)
\(\chi_{1620}(1519,\cdot)\)
\(\chi_{1620}(1579,\cdot)\)
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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 ]
\((811,1541,1297)\) → \((-1,e\left(\frac{10}{27}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1620 }(679, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{17}{27}\right)\) |
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sage:chi.jacobi_sum(n)
