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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4020, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,22,11,12]))
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pari:[g,chi] = znchar(Mod(107,4020))
| Modulus: | \(4020\) | |
| Conductor: | \(4020\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{4020}(107,\cdot)\)
\(\chi_{4020}(143,\cdot)\)
\(\chi_{4020}(263,\cdot)\)
\(\chi_{4020}(863,\cdot)\)
\(\chi_{4020}(947,\cdot)\)
\(\chi_{4020}(1067,\cdot)\)
\(\chi_{4020}(1163,\cdot)\)
\(\chi_{4020}(1667,\cdot)\)
\(\chi_{4020}(1823,\cdot)\)
\(\chi_{4020}(1967,\cdot)\)
\(\chi_{4020}(2303,\cdot)\)
\(\chi_{4020}(2543,\cdot)\)
\(\chi_{4020}(2627,\cdot)\)
\(\chi_{4020}(2903,\cdot)\)
\(\chi_{4020}(2963,\cdot)\)
\(\chi_{4020}(3107,\cdot)\)
\(\chi_{4020}(3323,\cdot)\)
\(\chi_{4020}(3347,\cdot)\)
\(\chi_{4020}(3707,\cdot)\)
\(\chi_{4020}(3767,\cdot)\)
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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 ]
\((2011,2681,3217,1141)\) → \((-1,-1,i,e\left(\frac{3}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4020 }(107, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(i\) | \(e\left(\frac{21}{22}\right)\) |
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sage:chi.jacobi_sum(n)
