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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4056, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,13,13,5]))
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pari:[g,chi] = znchar(Mod(155,4056))
| Modulus: | \(4056\) | |
| Conductor: | \(4056\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(26\) |
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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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\(\chi_{4056}(155,\cdot)\)
\(\chi_{4056}(467,\cdot)\)
\(\chi_{4056}(779,\cdot)\)
\(\chi_{4056}(1091,\cdot)\)
\(\chi_{4056}(1403,\cdot)\)
\(\chi_{4056}(1715,\cdot)\)
\(\chi_{4056}(2339,\cdot)\)
\(\chi_{4056}(2651,\cdot)\)
\(\chi_{4056}(2963,\cdot)\)
\(\chi_{4056}(3275,\cdot)\)
\(\chi_{4056}(3587,\cdot)\)
\(\chi_{4056}(3899,\cdot)\)
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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 ]
\((1015,2029,2705,3889)\) → \((-1,-1,-1,e\left(\frac{5}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4056 }(155, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) |
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sage:chi.jacobi_sum(n)
