from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(260, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([6,9,7]))
pari: [g,chi] = znchar(Mod(63,260))
Basic properties
| Modulus: | \(260\) | |
| Conductor: | \(260\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 260.be
\(\chi_{260}(63,\cdot)\) \(\chi_{260}(67,\cdot)\) \(\chi_{260}(163,\cdot)\) \(\chi_{260}(227,\cdot)\)
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: | 12.0.14337283152296000000000.2 |
Values on generators
\((131,157,41)\) → \((-1,-i,e\left(\frac{7}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 260 }(63, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) | \(i\) | \(e\left(\frac{5}{6}\right)\) |
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)