from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(955, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,5]))
pari: [g,chi] = znchar(Mod(14,955))
Basic properties
Modulus: | \(955\) | |
Conductor: | \(955\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 955.p
\(\chi_{955}(14,\cdot)\) \(\chi_{955}(84,\cdot)\) \(\chi_{955}(139,\cdot)\) \(\chi_{955}(159,\cdot)\) \(\chi_{955}(229,\cdot)\) \(\chi_{955}(419,\cdot)\) \(\chi_{955}(504,\cdot)\) \(\chi_{955}(584,\cdot)\) \(\chi_{955}(604,\cdot)\) \(\chi_{955}(614,\cdot)\) \(\chi_{955}(639,\cdot)\) \(\chi_{955}(734,\cdot)\) \(\chi_{955}(739,\cdot)\) \(\chi_{955}(759,\cdot)\) \(\chi_{955}(819,\cdot)\) \(\chi_{955}(834,\cdot)\) \(\chi_{955}(919,\cdot)\) \(\chi_{955}(949,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
Fixed field: | 38.0.47716070387122491878768794713669776106334434794159463718664766104756555702804321854228973388671875.1 |
Values on generators
\((192,401)\) → \((-1,e\left(\frac{5}{38}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 955 }(14, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(1\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{9}{38}\right)\) |
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)