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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7865, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([11,2,22]))
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pari:[g,chi] = znchar(Mod(7292,7865))
| Modulus: | \(7865\) | |
| Conductor: | \(7865\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(44\) |
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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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\(\chi_{7865}(142,\cdot)\)
\(\chi_{7865}(428,\cdot)\)
\(\chi_{7865}(857,\cdot)\)
\(\chi_{7865}(1143,\cdot)\)
\(\chi_{7865}(1858,\cdot)\)
\(\chi_{7865}(2287,\cdot)\)
\(\chi_{7865}(2573,\cdot)\)
\(\chi_{7865}(3002,\cdot)\)
\(\chi_{7865}(3288,\cdot)\)
\(\chi_{7865}(3717,\cdot)\)
\(\chi_{7865}(4003,\cdot)\)
\(\chi_{7865}(4432,\cdot)\)
\(\chi_{7865}(5147,\cdot)\)
\(\chi_{7865}(5433,\cdot)\)
\(\chi_{7865}(5862,\cdot)\)
\(\chi_{7865}(6148,\cdot)\)
\(\chi_{7865}(6577,\cdot)\)
\(\chi_{7865}(6863,\cdot)\)
\(\chi_{7865}(7292,\cdot)\)
\(\chi_{7865}(7578,\cdot)\)
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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 ]
\((3147,3511,1211)\) → \((i,e\left(\frac{1}{22}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(14\) | \(16\) |
| \( \chi_{ 7865 }(7292, a) \) |
\(1\) | \(1\) | \(e\left(\frac{35}{44}\right)\) | \(-i\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(-1\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) |
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sage:chi.jacobi_sum(n)
