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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([69,86]))
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pari:[g,chi] = znchar(Mod(5,417))
| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{417}(5,\cdot)\)
\(\chi_{417}(11,\cdot)\)
\(\chi_{417}(20,\cdot)\)
\(\chi_{417}(29,\cdot)\)
\(\chi_{417}(35,\cdot)\)
\(\chi_{417}(38,\cdot)\)
\(\chi_{417}(41,\cdot)\)
\(\chi_{417}(47,\cdot)\)
\(\chi_{417}(71,\cdot)\)
\(\chi_{417}(83,\cdot)\)
\(\chi_{417}(86,\cdot)\)
\(\chi_{417}(89,\cdot)\)
\(\chi_{417}(107,\cdot)\)
\(\chi_{417}(113,\cdot)\)
\(\chi_{417}(122,\cdot)\)
\(\chi_{417}(137,\cdot)\)
\(\chi_{417}(143,\cdot)\)
\(\chi_{417}(146,\cdot)\)
\(\chi_{417}(152,\cdot)\)
\(\chi_{417}(155,\cdot)\)
\(\chi_{417}(164,\cdot)\)
\(\chi_{417}(167,\cdot)\)
\(\chi_{417}(170,\cdot)\)
\(\chi_{417}(176,\cdot)\)
\(\chi_{417}(185,\cdot)\)
\(\chi_{417}(188,\cdot)\)
\(\chi_{417}(206,\cdot)\)
\(\chi_{417}(257,\cdot)\)
\(\chi_{417}(260,\cdot)\)
\(\chi_{417}(263,\cdot)\)
...
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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 ]
\((140,280)\) → \((-1,e\left(\frac{43}{69}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 417 }(5, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{17}{138}\right)\) | \(e\left(\frac{17}{69}\right)\) | \(e\left(\frac{13}{138}\right)\) | \(e\left(\frac{11}{69}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{119}{138}\right)\) | \(e\left(\frac{61}{69}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{34}{69}\right)\) |
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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)
