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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1960, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,21,41]))
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pari:[g,chi] = znchar(Mod(1699,1960))
| Modulus: | \(1960\) | |
| Conductor: | \(1960\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(42\) |
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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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\(\chi_{1960}(59,\cdot)\)
\(\chi_{1960}(299,\cdot)\)
\(\chi_{1960}(339,\cdot)\)
\(\chi_{1960}(579,\cdot)\)
\(\chi_{1960}(859,\cdot)\)
\(\chi_{1960}(899,\cdot)\)
\(\chi_{1960}(1139,\cdot)\)
\(\chi_{1960}(1179,\cdot)\)
\(\chi_{1960}(1419,\cdot)\)
\(\chi_{1960}(1459,\cdot)\)
\(\chi_{1960}(1699,\cdot)\)
\(\chi_{1960}(1739,\cdot)\)
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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 ]
\((1471,981,1177,1081)\) → \((-1,-1,-1,e\left(\frac{41}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 1960 }(1699, a) \) |
\(1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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sage:chi.jacobi_sum(n)
