sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(105)
sage: chi = H[13]
pari: [g,chi] = znchar(Mod(13,105))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 35 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 4 |
| Real | = | no |
| sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
| ||
| Primitive | = | no |
| Minimal | = | yes |
| sage: chi.is_odd()
pari: zncharisodd(g,chi)
| ||
| Parity | = | even |
| Orbit label | = | 105.m |
| Orbit index | = | 13 |
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 ]
\(\chi_{105}(13,\cdot)\) \(\chi_{105}(97,\cdot)\)
Values on generators
\((71,22,31)\) → \((1,-i,-1)\)
Values
| -1 | 1 | 2 | 4 | 8 | 11 | 13 | 16 | 17 | 19 | 22 | 23 |
| \(1\) | \(1\) | \(-i\) | \(-1\) | \(i\) | \(1\) | \(-i\) | \(1\) | \(i\) | \(1\) | \(-i\) | \(i\) |
Related number fields
| Field of values | \(\Q(i)\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{105}(13,\cdot)) = \sum_{r\in \Z/105\Z} \chi_{105}(13,r) e\left(\frac{2r}{105}\right) = 3.1102672038+5.0325180498i \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{105}(13,\cdot),\chi_{105}(1,\cdot)) = \sum_{r\in \Z/105\Z} \chi_{105}(13,r) \chi_{105}(1,1-r) = 1 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{105}(13,·))
= \sum_{r \in \Z/105\Z}
\chi_{105}(13,r) e\left(\frac{1 r + 2 r^{-1}}{105}\right)
= 7.7568795304+-7.7568795304i \)