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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(669, base_ring=CyclotomicField(74))
M = H._module
chi = DirichletCharacter(H, M([37,32]))
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pari:[g,chi] = znchar(Mod(8,669))
| Modulus: | \(669\) | |
| Conductor: | \(669\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(74\) |
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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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\(\chi_{669}(2,\cdot)\)
\(\chi_{669}(8,\cdot)\)
\(\chi_{669}(14,\cdot)\)
\(\chi_{669}(17,\cdot)\)
\(\chi_{669}(32,\cdot)\)
\(\chi_{669}(41,\cdot)\)
\(\chi_{669}(56,\cdot)\)
\(\chi_{669}(68,\cdot)\)
\(\chi_{669}(98,\cdot)\)
\(\chi_{669}(119,\cdot)\)
\(\chi_{669}(128,\cdot)\)
\(\chi_{669}(164,\cdot)\)
\(\chi_{669}(197,\cdot)\)
\(\chi_{669}(227,\cdot)\)
\(\chi_{669}(230,\cdot)\)
\(\chi_{669}(239,\cdot)\)
\(\chi_{669}(251,\cdot)\)
\(\chi_{669}(257,\cdot)\)
\(\chi_{669}(272,\cdot)\)
\(\chi_{669}(287,\cdot)\)
\(\chi_{669}(305,\cdot)\)
\(\chi_{669}(335,\cdot)\)
\(\chi_{669}(338,\cdot)\)
\(\chi_{669}(359,\cdot)\)
\(\chi_{669}(392,\cdot)\)
\(\chi_{669}(419,\cdot)\)
\(\chi_{669}(461,\cdot)\)
\(\chi_{669}(476,\cdot)\)
\(\chi_{669}(479,\cdot)\)
\(\chi_{669}(506,\cdot)\)
...
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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 ]
\((224,226)\) → \((-1,e\left(\frac{16}{37}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 669 }(8, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{25}{74}\right)\) | \(e\left(\frac{25}{37}\right)\) | \(e\left(\frac{73}{74}\right)\) | \(e\left(\frac{30}{37}\right)\) | \(e\left(\frac{1}{74}\right)\) | \(e\left(\frac{12}{37}\right)\) | \(e\left(\frac{57}{74}\right)\) | \(e\left(\frac{21}{37}\right)\) | \(e\left(\frac{11}{74}\right)\) | \(e\left(\frac{13}{37}\right)\) |
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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)
