from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(967, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([38]))
pari: [g,chi] = znchar(Mod(703,967))
Basic properties
| Modulus: | \(967\) | |
| Conductor: | \(967\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(23\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 967.h
\(\chi_{967}(69,\cdot)\) \(\chi_{967}(72,\cdot)\) \(\chi_{967}(133,\cdot)\) \(\chi_{967}(157,\cdot)\) \(\chi_{967}(187,\cdot)\) \(\chi_{967}(196,\cdot)\) \(\chi_{967}(283,\cdot)\) \(\chi_{967}(332,\cdot)\) \(\chi_{967}(349,\cdot)\) \(\chi_{967}(474,\cdot)\) \(\chi_{967}(574,\cdot)\) \(\chi_{967}(641,\cdot)\) \(\chi_{967}(667,\cdot)\) \(\chi_{967}(696,\cdot)\) \(\chi_{967}(703,\cdot)\) \(\chi_{967}(714,\cdot)\) \(\chi_{967}(795,\cdot)\) \(\chi_{967}(873,\cdot)\) \(\chi_{967}(893,\cdot)\) \(\chi_{967}(916,\cdot)\) \(\chi_{967}(926,\cdot)\) \(\chi_{967}(953,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{23})\) |
| Fixed field: | Number field defined by a degree 23 polynomial |
Values on generators
\(5\) → \(e\left(\frac{19}{23}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 967 }(703, a) \) | \(1\) | \(1\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)