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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3724, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,5,14]))
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pari:[g,chi] = znchar(Mod(1223,3724))
| Modulus: | \(3724\) | |
| Conductor: | \(3724\) |
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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)
|
| 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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\(\chi_{3724}(87,\cdot)\)
\(\chi_{3724}(159,\cdot)\)
\(\chi_{3724}(691,\cdot)\)
\(\chi_{3724}(1151,\cdot)\)
\(\chi_{3724}(1223,\cdot)\)
\(\chi_{3724}(1683,\cdot)\)
\(\chi_{3724}(1755,\cdot)\)
\(\chi_{3724}(2215,\cdot)\)
\(\chi_{3724}(2287,\cdot)\)
\(\chi_{3724}(2747,\cdot)\)
\(\chi_{3724}(2819,\cdot)\)
\(\chi_{3724}(3279,\cdot)\)
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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 ]
\((1863,3041,3137)\) → \((-1,e\left(\frac{5}{42}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 3724 }(1223, a) \) |
\(1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) |
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sage:chi.jacobi_sum(n)
