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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(223, base_ring=CyclotomicField(74))
M = H._module
chi = DirichletCharacter(H, M([18]))
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pari:[g,chi] = znchar(Mod(16,223))
| Modulus: | \(223\) | |
| Conductor: | \(223\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(37\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{223}(2,\cdot)\)
\(\chi_{223}(4,\cdot)\)
\(\chi_{223}(7,\cdot)\)
\(\chi_{223}(8,\cdot)\)
\(\chi_{223}(14,\cdot)\)
\(\chi_{223}(15,\cdot)\)
\(\chi_{223}(16,\cdot)\)
\(\chi_{223}(17,\cdot)\)
\(\chi_{223}(28,\cdot)\)
\(\chi_{223}(30,\cdot)\)
\(\chi_{223}(32,\cdot)\)
\(\chi_{223}(33,\cdot)\)
\(\chi_{223}(34,\cdot)\)
\(\chi_{223}(41,\cdot)\)
\(\chi_{223}(49,\cdot)\)
\(\chi_{223}(56,\cdot)\)
\(\chi_{223}(60,\cdot)\)
\(\chi_{223}(64,\cdot)\)
\(\chi_{223}(66,\cdot)\)
\(\chi_{223}(68,\cdot)\)
\(\chi_{223}(82,\cdot)\)
\(\chi_{223}(98,\cdot)\)
\(\chi_{223}(105,\cdot)\)
\(\chi_{223}(112,\cdot)\)
\(\chi_{223}(115,\cdot)\)
\(\chi_{223}(119,\cdot)\)
\(\chi_{223}(120,\cdot)\)
\(\chi_{223}(128,\cdot)\)
\(\chi_{223}(132,\cdot)\)
\(\chi_{223}(136,\cdot)\)
...
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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 ]
\(3\) → \(e\left(\frac{9}{37}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 223 }(16, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{37}\right)\) | \(e\left(\frac{9}{37}\right)\) | \(e\left(\frac{21}{37}\right)\) | \(e\left(\frac{24}{37}\right)\) | \(e\left(\frac{1}{37}\right)\) | \(e\left(\frac{3}{37}\right)\) | \(e\left(\frac{13}{37}\right)\) | \(e\left(\frac{18}{37}\right)\) | \(e\left(\frac{16}{37}\right)\) | \(e\left(\frac{1}{37}\right)\) |
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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)
