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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4056, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([39,39,39,35]))
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pari:[g,chi] = znchar(Mod(1115,4056))
| Modulus: | \(4056\) | |
| Conductor: | \(4056\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(78\) |
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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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\(\chi_{4056}(179,\cdot)\)
\(\chi_{4056}(251,\cdot)\)
\(\chi_{4056}(491,\cdot)\)
\(\chi_{4056}(563,\cdot)\)
\(\chi_{4056}(803,\cdot)\)
\(\chi_{4056}(875,\cdot)\)
\(\chi_{4056}(1115,\cdot)\)
\(\chi_{4056}(1187,\cdot)\)
\(\chi_{4056}(1427,\cdot)\)
\(\chi_{4056}(1739,\cdot)\)
\(\chi_{4056}(1811,\cdot)\)
\(\chi_{4056}(2123,\cdot)\)
\(\chi_{4056}(2363,\cdot)\)
\(\chi_{4056}(2435,\cdot)\)
\(\chi_{4056}(2675,\cdot)\)
\(\chi_{4056}(2747,\cdot)\)
\(\chi_{4056}(2987,\cdot)\)
\(\chi_{4056}(3059,\cdot)\)
\(\chi_{4056}(3299,\cdot)\)
\(\chi_{4056}(3371,\cdot)\)
\(\chi_{4056}(3611,\cdot)\)
\(\chi_{4056}(3683,\cdot)\)
\(\chi_{4056}(3923,\cdot)\)
\(\chi_{4056}(3995,\cdot)\)
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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 ]
\((1015,2029,2705,3889)\) → \((-1,-1,-1,e\left(\frac{35}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4056 }(1115, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{43}{78}\right)\) |
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sage:chi.jacobi_sum(n)
