sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(65)
sage: chi = H[16]
pari: [g,chi] = znchar(Mod(16,65))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 13 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 3 |
| 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 | = | 65.e |
| Orbit index | = | 5 |
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 ]
\(\chi_{65}(16,\cdot)\) \(\chi_{65}(61,\cdot)\)
Values on generators
\((27,41)\) → \((1,e\left(\frac{1}{3}\right))\)
Values
| -1 | 1 | 2 | 3 | 4 | 6 | 7 | 8 | 9 | 11 | 12 | 14 |
| \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(1\) |
Related number fields
| Field of values | \(\Q(\zeta_{3})\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{65}(16,\cdot)) = \sum_{r\in \Z/65\Z} \chi_{65}(16,r) e\left(\frac{2r}{65}\right) = -2.565804281+2.5331104184i \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{65}(16,\cdot),\chi_{65}(1,\cdot)) = \sum_{r\in \Z/65\Z} \chi_{65}(16,r) \chi_{65}(1,1-r) = -3 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{65}(16,·))
= \sum_{r \in \Z/65\Z}
\chi_{65}(16,r) e\left(\frac{1 r + 2 r^{-1}}{65}\right)
= 1.4986674202+2.5957681155i \)