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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4056, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([0,78,78,37]))
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pari:[g,chi] = znchar(Mod(1757,4056))
| Modulus: | \(4056\) | |
| Conductor: | \(4056\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(156\) |
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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)
|
\(\chi_{4056}(149,\cdot)\)
\(\chi_{4056}(197,\cdot)\)
\(\chi_{4056}(245,\cdot)\)
\(\chi_{4056}(293,\cdot)\)
\(\chi_{4056}(461,\cdot)\)
\(\chi_{4056}(509,\cdot)\)
\(\chi_{4056}(557,\cdot)\)
\(\chi_{4056}(605,\cdot)\)
\(\chi_{4056}(773,\cdot)\)
\(\chi_{4056}(821,\cdot)\)
\(\chi_{4056}(869,\cdot)\)
\(\chi_{4056}(917,\cdot)\)
\(\chi_{4056}(1085,\cdot)\)
\(\chi_{4056}(1133,\cdot)\)
\(\chi_{4056}(1181,\cdot)\)
\(\chi_{4056}(1229,\cdot)\)
\(\chi_{4056}(1397,\cdot)\)
\(\chi_{4056}(1445,\cdot)\)
\(\chi_{4056}(1493,\cdot)\)
\(\chi_{4056}(1541,\cdot)\)
\(\chi_{4056}(1757,\cdot)\)
\(\chi_{4056}(1805,\cdot)\)
\(\chi_{4056}(1853,\cdot)\)
\(\chi_{4056}(2021,\cdot)\)
\(\chi_{4056}(2069,\cdot)\)
\(\chi_{4056}(2165,\cdot)\)
\(\chi_{4056}(2333,\cdot)\)
\(\chi_{4056}(2381,\cdot)\)
\(\chi_{4056}(2429,\cdot)\)
\(\chi_{4056}(2477,\cdot)\)
...
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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 ]
\((1015,2029,2705,3889)\) → \((1,-1,-1,e\left(\frac{37}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4056 }(1757, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{59}{156}\right)\) | \(e\left(\frac{67}{156}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{20}{39}\right)\) |
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sage:chi.jacobi_sum(n)
