sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(7)
sage: chi = H[6]
pari: [g,chi] = znchar(Mod(6,7))
Kronecker symbol representation
sage: kronecker_character(-7)
pari: znchartokronecker(g,chi)
\(\displaystyle\left(\frac{-7}{\bullet}\right)\)
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 7 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 2 |
| Real | = | yes |
| sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
| ||
| Primitive | = | yes |
| Minimal | = | yes |
| sage: chi.is_odd()
pari: zncharisodd(g,chi)
| ||
| Parity | = | odd |
| Orbit label | = | 7.b |
| Orbit index | = | 2 |
Galois orbit
sage: chi.sage_character().galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Values on generators
\(3\) → \(-1\)
Values
| -1 | 1 | 2 | 3 | 4 | 5 |
| \(-1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) |
Related number fields
| Field of values | \(\Q\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{7}(6,\cdot)) = \sum_{r\in \Z/7\Z} \chi_{7}(6,r) e\left(\frac{2r}{7}\right) = 2.6457513111i \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{7}(6,\cdot),\chi_{7}(1,\cdot)) = \sum_{r\in \Z/7\Z} \chi_{7}(6,r) \chi_{7}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{7}(6,·))
= \sum_{r \in \Z/7\Z}
\chi_{7}(6,r) e\left(\frac{1 r + 2 r^{-1}}{7}\right)
= 3.2991979214i \)