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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(121, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([87]))
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pari:[g,chi] = znchar(Mod(62,121))
| Modulus: | \(121\) | |
| Conductor: | \(121\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(110\) |
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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
|
| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{121}(2,\cdot)\)
\(\chi_{121}(6,\cdot)\)
\(\chi_{121}(7,\cdot)\)
\(\chi_{121}(8,\cdot)\)
\(\chi_{121}(13,\cdot)\)
\(\chi_{121}(17,\cdot)\)
\(\chi_{121}(18,\cdot)\)
\(\chi_{121}(19,\cdot)\)
\(\chi_{121}(24,\cdot)\)
\(\chi_{121}(28,\cdot)\)
\(\chi_{121}(29,\cdot)\)
\(\chi_{121}(30,\cdot)\)
\(\chi_{121}(35,\cdot)\)
\(\chi_{121}(39,\cdot)\)
\(\chi_{121}(41,\cdot)\)
\(\chi_{121}(46,\cdot)\)
\(\chi_{121}(50,\cdot)\)
\(\chi_{121}(51,\cdot)\)
\(\chi_{121}(52,\cdot)\)
\(\chi_{121}(57,\cdot)\)
\(\chi_{121}(61,\cdot)\)
\(\chi_{121}(62,\cdot)\)
\(\chi_{121}(63,\cdot)\)
\(\chi_{121}(68,\cdot)\)
\(\chi_{121}(72,\cdot)\)
\(\chi_{121}(73,\cdot)\)
\(\chi_{121}(74,\cdot)\)
\(\chi_{121}(79,\cdot)\)
\(\chi_{121}(83,\cdot)\)
\(\chi_{121}(84,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{87}{110}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 121 }(62, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) |
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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)
