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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([0,13,13,4]))
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pari:[g,chi] = znchar(Mod(3797,4056))
| 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
|
| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{4056}(53,\cdot)\)
\(\chi_{4056}(365,\cdot)\)
\(\chi_{4056}(989,\cdot)\)
\(\chi_{4056}(1301,\cdot)\)
\(\chi_{4056}(1613,\cdot)\)
\(\chi_{4056}(1925,\cdot)\)
\(\chi_{4056}(2237,\cdot)\)
\(\chi_{4056}(2549,\cdot)\)
\(\chi_{4056}(2861,\cdot)\)
\(\chi_{4056}(3173,\cdot)\)
\(\chi_{4056}(3485,\cdot)\)
\(\chi_{4056}(3797,\cdot)\)
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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 ]
\((1015,2029,2705,3889)\) → \((1,-1,-1,e\left(\frac{2}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4056 }(3797, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) |
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sage:chi.jacobi_sum(n)
