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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([37,27]))
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pari:[g,chi] = znchar(Mod(605,669))
| 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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{669}(26,\cdot)\)
\(\chi_{669}(59,\cdot)\)
\(\chi_{669}(95,\cdot)\)
\(\chi_{669}(104,\cdot)\)
\(\chi_{669}(125,\cdot)\)
\(\chi_{669}(155,\cdot)\)
\(\chi_{669}(167,\cdot)\)
\(\chi_{669}(182,\cdot)\)
\(\chi_{669}(191,\cdot)\)
\(\chi_{669}(206,\cdot)\)
\(\chi_{669}(209,\cdot)\)
\(\chi_{669}(215,\cdot)\)
\(\chi_{669}(221,\cdot)\)
\(\chi_{669}(236,\cdot)\)
\(\chi_{669}(275,\cdot)\)
\(\chi_{669}(314,\cdot)\)
\(\chi_{669}(326,\cdot)\)
\(\chi_{669}(341,\cdot)\)
\(\chi_{669}(380,\cdot)\)
\(\chi_{669}(386,\cdot)\)
\(\chi_{669}(413,\cdot)\)
\(\chi_{669}(416,\cdot)\)
\(\chi_{669}(431,\cdot)\)
\(\chi_{669}(473,\cdot)\)
\(\chi_{669}(500,\cdot)\)
\(\chi_{669}(533,\cdot)\)
\(\chi_{669}(554,\cdot)\)
\(\chi_{669}(557,\cdot)\)
\(\chi_{669}(587,\cdot)\)
\(\chi_{669}(605,\cdot)\)
...
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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 ]
\((224,226)\) → \((-1,e\left(\frac{27}{74}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 669 }(605, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{74}\right)\) | \(e\left(\frac{13}{37}\right)\) | \(e\left(\frac{36}{37}\right)\) | \(e\left(\frac{23}{37}\right)\) | \(e\left(\frac{39}{74}\right)\) | \(e\left(\frac{11}{74}\right)\) | \(e\left(\frac{20}{37}\right)\) | \(e\left(\frac{47}{74}\right)\) | \(e\left(\frac{59}{74}\right)\) | \(e\left(\frac{26}{37}\right)\) |
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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)
