sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(35)
sage: chi = H[3]
pari: [g,chi] = znchar(Mod(3,35))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 35 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 12 |
| 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 | = | Even |
| Orbit label | = | 35.k |
| Orbit index | = | 11 |
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 ]
\(\chi_{35}(3,\cdot)\) \(\chi_{35}(12,\cdot)\) \(\chi_{35}(17,\cdot)\) \(\chi_{35}(33,\cdot)\)
Values on generators
\((22,31)\) → \((-i,e\left(\frac{1}{6}\right))\)
Values
| -1 | 1 | 2 | 3 | 4 | 6 | 8 | 9 | 11 | 12 | 13 | 16 |
| \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(i\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(-i\) | \(e\left(\frac{1}{3}\right)\) |
Related number fields
| Field of values | \(\Q(\zeta_{12})\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{35}(3,\cdot)) = \sum_{r\in \Z/35\Z} \chi_{35}(3,r) e\left(\frac{2r}{35}\right) = 4.8136853347+3.439248973i \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{35}(3,\cdot),\chi_{35}(1,\cdot)) = \sum_{r\in \Z/35\Z} \chi_{35}(3,r) \chi_{35}(1,1-r) = 1 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{35}(3,·))
= \sum_{r \in \Z/35\Z}
\chi_{35}(3,r) e\left(\frac{1 r + 2 r^{-1}}{35}\right)
= 2.8116277235+0.7533733779i \)