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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(828, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,22,18]))
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pari:[g,chi] = znchar(Mod(31,828))
| Modulus: | \(828\) | |
| Conductor: | \(828\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(66\) |
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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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\(\chi_{828}(31,\cdot)\)
\(\chi_{828}(151,\cdot)\)
\(\chi_{828}(187,\cdot)\)
\(\chi_{828}(211,\cdot)\)
\(\chi_{828}(223,\cdot)\)
\(\chi_{828}(259,\cdot)\)
\(\chi_{828}(331,\cdot)\)
\(\chi_{828}(403,\cdot)\)
\(\chi_{828}(427,\cdot)\)
\(\chi_{828}(439,\cdot)\)
\(\chi_{828}(463,\cdot)\)
\(\chi_{828}(499,\cdot)\)
\(\chi_{828}(535,\cdot)\)
\(\chi_{828}(547,\cdot)\)
\(\chi_{828}(583,\cdot)\)
\(\chi_{828}(607,\cdot)\)
\(\chi_{828}(679,\cdot)\)
\(\chi_{828}(715,\cdot)\)
\(\chi_{828}(763,\cdot)\)
\(\chi_{828}(823,\cdot)\)
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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 ]
\((415,461,649)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{3}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 828 }(31, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
