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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([30,51,50]))
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pari:[g,chi] = znchar(Mod(647,975))
| 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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\(\chi_{975}(17,\cdot)\)
\(\chi_{975}(23,\cdot)\)
\(\chi_{975}(62,\cdot)\)
\(\chi_{975}(173,\cdot)\)
\(\chi_{975}(212,\cdot)\)
\(\chi_{975}(413,\cdot)\)
\(\chi_{975}(452,\cdot)\)
\(\chi_{975}(563,\cdot)\)
\(\chi_{975}(602,\cdot)\)
\(\chi_{975}(608,\cdot)\)
\(\chi_{975}(647,\cdot)\)
\(\chi_{975}(758,\cdot)\)
\(\chi_{975}(797,\cdot)\)
\(\chi_{975}(803,\cdot)\)
\(\chi_{975}(842,\cdot)\)
\(\chi_{975}(953,\cdot)\)
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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 ]
\((326,352,301)\) → \((-1,e\left(\frac{17}{20}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 975 }(647, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) |
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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)
