sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(11)
sage: chi = H[10]
sage: H = DirichletGroup_conrey(11)
sage: chi = H[10]
pari: [g,chi] = znchar(Mod(10,11))
Kronecker symbol representation
sage: kronecker_character(-11)
\(\displaystyle\left(\frac{-11}{\bullet}\right)\)
Values on generators
sage: chi(k) for k in H.gens()
pari: [ chareval(g,chi,x) | x <- g.gen ] \\ value in Q/Z
\(2\) → \(-1\)
Values
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 1 | \(-1\) | 1 | 1 | 1 | \(-1\) | \(-1\) | \(-1\) | 1 | \(-1\) |
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 11 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 2 |
| sage: chi.is_odd()
pari: zncharisodd(g,chi)
| ||
| Parity | = | Odd |
| Real | = | Yes |
| sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
| ||
| Primitive | = | Yes |
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 ]
Related number fields
| Field of values | \(\Q\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{11}(10,\cdot)) = \sum_{r\in \Z/11\Z} \chi_{11}(10,r) e\left(\frac{2r}{11}\right) = -3.3166247904i. \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{11}(10,\cdot),\chi_{11}(1,\cdot)) = \sum_{r\in \Z/11\Z} \chi_{11}(10,r) \chi_{11}(1,1-r) = -1.\)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{11}(10,·))
= \sum_{r \in \Z/11\Z}
\chi_{11}(10,r) e\left(\frac{1 r + 2 r^{-1}}{11}\right)
= -0.0. \)