sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(28)
sage: chi = H[23]
pari: [g,chi] = znchar(Mod(23,28))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 28 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 6 |
| Real | = | No |
| sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
| ||
| Primitive | = | Yes |
| sage: chi.is_odd()
pari: zncharisodd(g,chi)
| ||
| Parity | = | Odd |
| Orbit label | = | 28.g |
| Orbit index | = | 7 |
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 ]
\(\chi_{28}(11,\cdot)\) \(\chi_{28}(23,\cdot)\)
Values on generators
\((15,17)\) → \((-1,e\left(\frac{1}{3}\right))\)
Values
| -1 | 1 | 3 | 5 | 9 | 11 | 13 | 15 | 17 | 19 | 23 | 25 |
| \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(1\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) |
Related number fields
| Field of values | \(\Q(\zeta_{3})\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{28}(23,\cdot)) = \sum_{r\in \Z/28\Z} \chi_{28}(23,r) e\left(\frac{r}{14}\right) = -0.0 \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{28}(23,\cdot),\chi_{28}(1,\cdot)) = \sum_{r\in \Z/28\Z} \chi_{28}(23,r) \chi_{28}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{28}(23,·))
= \sum_{r \in \Z/28\Z}
\chi_{28}(23,r) e\left(\frac{1 r + 2 r^{-1}}{28}\right)
= 2.9306671564+1.6920214716i \)