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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(975, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,3,20]))
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pari:[g,chi] = znchar(Mod(302,975))
| Modulus: | \(975\) | |
| Conductor: | \(975\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(60\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{975}(113,\cdot)\)
\(\chi_{975}(152,\cdot)\)
\(\chi_{975}(263,\cdot)\)
\(\chi_{975}(302,\cdot)\)
\(\chi_{975}(308,\cdot)\)
\(\chi_{975}(347,\cdot)\)
\(\chi_{975}(458,\cdot)\)
\(\chi_{975}(497,\cdot)\)
\(\chi_{975}(503,\cdot)\)
\(\chi_{975}(542,\cdot)\)
\(\chi_{975}(653,\cdot)\)
\(\chi_{975}(692,\cdot)\)
\(\chi_{975}(698,\cdot)\)
\(\chi_{975}(737,\cdot)\)
\(\chi_{975}(848,\cdot)\)
\(\chi_{975}(887,\cdot)\)
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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 ]
\((326,352,301)\) → \((-1,e\left(\frac{1}{20}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 975 }(302, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) |
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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)
