sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(29)
sage: chi = H[3]
pari: [g,chi] = znchar(Mod(3,29))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 29 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 28 |
| 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 | = | Odd |
| Orbit label | = | 29.f |
| Orbit index | = | 6 |
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 ]
\(\chi_{29}(2,\cdot)\) \(\chi_{29}(3,\cdot)\) \(\chi_{29}(8,\cdot)\) \(\chi_{29}(10,\cdot)\) \(\chi_{29}(11,\cdot)\) \(\chi_{29}(14,\cdot)\) \(\chi_{29}(15,\cdot)\) \(\chi_{29}(18,\cdot)\) \(\chi_{29}(19,\cdot)\) \(\chi_{29}(21,\cdot)\) \(\chi_{29}(26,\cdot)\) \(\chi_{29}(27,\cdot)\)
Values on generators
\(2\) → \(e\left(\frac{5}{28}\right)\)
Values
| -1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| \(-1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) |
Related number fields
| Field of values | \(\Q(\zeta_{28})\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{29}(3,\cdot)) = \sum_{r\in \Z/29\Z} \chi_{29}(3,r) e\left(\frac{2r}{29}\right) = 1.8410703114+5.0606778309i \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{29}(3,\cdot),\chi_{29}(1,\cdot)) = \sum_{r\in \Z/29\Z} \chi_{29}(3,r) \chi_{29}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{29}(3,·))
= \sum_{r \in \Z/29\Z}
\chi_{29}(3,r) e\left(\frac{1 r + 2 r^{-1}}{29}\right)
= 0.4484787206+-0.713749795i \)