sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(10)
sage: chi = H[3]
sage: H = DirichletGroup_conrey(10)
sage: chi = H[3]
pari: [g,chi] = znchar(Mod(3,10))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 5 |
| 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 |
| sage: chi.is_odd()
pari: zncharisodd(g,chi)
| ||
| Parity | = | Odd |
| Orbit label | = | 10.c |
| Orbit index | = | 3 |
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 ]
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\(\chi_{10}(3,\cdot)\) \(\chi_{10}(7,\cdot)\)
Inducing primitive character
Values on generators
\(7\) → \(-i\)
Values
| -1 | 1 | 3 | 7 |
| \(-1\) | \(1\) | \(i\) | \(-i\) |
Related number fields
| Field of values | \(\Q(i)\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{10}(3,\cdot)) = \sum_{r\in \Z/10\Z} \chi_{10}(3,r) e\left(\frac{r}{5}\right) = 1.1755705046+1.9021130326i \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{10}(3,\cdot),\chi_{10}(1,\cdot)) = \sum_{r\in \Z/10\Z} \chi_{10}(3,r) \chi_{10}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{10}(3,·))
= \sum_{r \in \Z/10\Z}
\chi_{10}(3,r) e\left(\frac{1 r + 2 r^{-1}}{10}\right)
= 1.9021130326+1.9021130326i \)