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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(417, base_ring=CyclotomicField(138))
M = H._module
chi = DirichletCharacter(H, M([69,131]))
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pari:[g,chi] = znchar(Mod(101,417))
| Modulus: | \(417\) | |
| Conductor: | \(417\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(138\) |
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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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\(\chi_{417}(2,\cdot)\)
\(\chi_{417}(17,\cdot)\)
\(\chi_{417}(26,\cdot)\)
\(\chi_{417}(32,\cdot)\)
\(\chi_{417}(50,\cdot)\)
\(\chi_{417}(53,\cdot)\)
\(\chi_{417}(56,\cdot)\)
\(\chi_{417}(68,\cdot)\)
\(\chi_{417}(92,\cdot)\)
\(\chi_{417}(98,\cdot)\)
\(\chi_{417}(101,\cdot)\)
\(\chi_{417}(104,\cdot)\)
\(\chi_{417}(110,\cdot)\)
\(\chi_{417}(119,\cdot)\)
\(\chi_{417}(128,\cdot)\)
\(\chi_{417}(134,\cdot)\)
\(\chi_{417}(158,\cdot)\)
\(\chi_{417}(161,\cdot)\)
\(\chi_{417}(179,\cdot)\)
\(\chi_{417}(197,\cdot)\)
\(\chi_{417}(200,\cdot)\)
\(\chi_{417}(209,\cdot)\)
\(\chi_{417}(212,\cdot)\)
\(\chi_{417}(224,\cdot)\)
\(\chi_{417}(227,\cdot)\)
\(\chi_{417}(248,\cdot)\)
\(\chi_{417}(254,\cdot)\)
\(\chi_{417}(269,\cdot)\)
\(\chi_{417}(281,\cdot)\)
\(\chi_{417}(290,\cdot)\)
...
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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 ]
\((140,280)\) → \((-1,e\left(\frac{131}{138}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 417 }(101, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{69}\right)\) | \(e\left(\frac{62}{69}\right)\) | \(e\left(\frac{19}{138}\right)\) | \(e\left(\frac{32}{69}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{89}{138}\right)\) | \(e\left(\frac{52}{69}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{55}{69}\right)\) |
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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)
