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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4356, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,22,57]))
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pari:[g,chi] = znchar(Mod(4135,4356))
| Modulus: | \(4356\) | |
| Conductor: | \(4356\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(66\) |
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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
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{4356}(43,\cdot)\)
\(\chi_{4356}(175,\cdot)\)
\(\chi_{4356}(439,\cdot)\)
\(\chi_{4356}(571,\cdot)\)
\(\chi_{4356}(835,\cdot)\)
\(\chi_{4356}(1231,\cdot)\)
\(\chi_{4356}(1363,\cdot)\)
\(\chi_{4356}(1627,\cdot)\)
\(\chi_{4356}(1759,\cdot)\)
\(\chi_{4356}(2023,\cdot)\)
\(\chi_{4356}(2155,\cdot)\)
\(\chi_{4356}(2551,\cdot)\)
\(\chi_{4356}(2815,\cdot)\)
\(\chi_{4356}(2947,\cdot)\)
\(\chi_{4356}(3211,\cdot)\)
\(\chi_{4356}(3343,\cdot)\)
\(\chi_{4356}(3607,\cdot)\)
\(\chi_{4356}(3739,\cdot)\)
\(\chi_{4356}(4003,\cdot)\)
\(\chi_{4356}(4135,\cdot)\)
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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 ]
\((2179,1937,1333)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{19}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4356 }(4135, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) |
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sage:chi.jacobi_sum(n)
