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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(405, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([47,27]))
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pari:[g,chi] = znchar(Mod(374,405))
| Modulus: | \(405\) | |
| Conductor: | \(405\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(54\) |
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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
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| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{405}(14,\cdot)\)
\(\chi_{405}(29,\cdot)\)
\(\chi_{405}(59,\cdot)\)
\(\chi_{405}(74,\cdot)\)
\(\chi_{405}(104,\cdot)\)
\(\chi_{405}(119,\cdot)\)
\(\chi_{405}(149,\cdot)\)
\(\chi_{405}(164,\cdot)\)
\(\chi_{405}(194,\cdot)\)
\(\chi_{405}(209,\cdot)\)
\(\chi_{405}(239,\cdot)\)
\(\chi_{405}(254,\cdot)\)
\(\chi_{405}(284,\cdot)\)
\(\chi_{405}(299,\cdot)\)
\(\chi_{405}(329,\cdot)\)
\(\chi_{405}(344,\cdot)\)
\(\chi_{405}(374,\cdot)\)
\(\chi_{405}(389,\cdot)\)
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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 ]
\((326,82)\) → \((e\left(\frac{47}{54}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 405 }(374, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) |
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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)
