from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(760, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([6,6,9,8]))
pari: [g,chi] = znchar(Mod(163,760))
Basic properties
| Modulus: | \(760\) | |
| Conductor: | \(760\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | 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 760.bw
\(\chi_{760}(83,\cdot)\) \(\chi_{760}(163,\cdot)\) \(\chi_{760}(387,\cdot)\) \(\chi_{760}(467,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
| Fixed field: | 12.12.8695584276992000000000.1 |
Values on generators
\((191,381,457,401)\) → \((-1,-1,-i,e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 760 }(163, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(i\) | \(e\left(\frac{5}{6}\right)\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(-i\) | \(e\left(\frac{1}{3}\right)\) |
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)