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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3332, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,40,21]))
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pari:[g,chi] = znchar(Mod(1971,3332))
| Modulus: | \(3332\) | |
| Conductor: | \(3332\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{3332}(135,\cdot)\)
\(\chi_{3332}(543,\cdot)\)
\(\chi_{3332}(611,\cdot)\)
\(\chi_{3332}(1019,\cdot)\)
\(\chi_{3332}(1087,\cdot)\)
\(\chi_{3332}(1495,\cdot)\)
\(\chi_{3332}(1563,\cdot)\)
\(\chi_{3332}(1971,\cdot)\)
\(\chi_{3332}(2447,\cdot)\)
\(\chi_{3332}(2515,\cdot)\)
\(\chi_{3332}(2923,\cdot)\)
\(\chi_{3332}(2991,\cdot)\)
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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 ]
\((1667,885,785)\) → \((-1,e\left(\frac{20}{21}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 3332 }(1971, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) |
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sage:chi.jacobi_sum(n)
