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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([21,10,21]))
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pari:[g,chi] = znchar(Mod(151,3724))
| 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)
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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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\(\chi_{3724}(151,\cdot)\)
\(\chi_{3724}(303,\cdot)\)
\(\chi_{3724}(683,\cdot)\)
\(\chi_{3724}(835,\cdot)\)
\(\chi_{3724}(1215,\cdot)\)
\(\chi_{3724}(1367,\cdot)\)
\(\chi_{3724}(1747,\cdot)\)
\(\chi_{3724}(1899,\cdot)\)
\(\chi_{3724}(2279,\cdot)\)
\(\chi_{3724}(2963,\cdot)\)
\(\chi_{3724}(3343,\cdot)\)
\(\chi_{3724}(3495,\cdot)\)
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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 ]
\((1863,3041,3137)\) → \((-1,e\left(\frac{5}{21}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 3724 }(151, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) |
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sage:chi.jacobi_sum(n)
