sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(82)
sage: chi = H[45]
pari: [g,chi] = znchar(Mod(45,82))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 41 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 10 |
| 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 | = | 82.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 ]
\(\chi_{82}(23,\cdot)\) \(\chi_{82}(25,\cdot)\) \(\chi_{82}(31,\cdot)\) \(\chi_{82}(45,\cdot)\)
Values on generators
\(47\) → \(e\left(\frac{3}{10}\right)\)
Values
| -1 | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 |
| \(1\) | \(1\) | \(-1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) |
Related number fields
| Field of values | \(\Q(\zeta_{5})\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{82}(45,\cdot)) = \sum_{r\in \Z/82\Z} \chi_{82}(45,r) e\left(\frac{r}{41}\right) = -5.4200035579+-3.4093344559i \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{82}(45,\cdot),\chi_{82}(1,\cdot)) = \sum_{r\in \Z/82\Z} \chi_{82}(45,r) \chi_{82}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{82}(45,·))
= \sum_{r \in \Z/82\Z}
\chi_{82}(45,r) e\left(\frac{1 r + 2 r^{-1}}{82}\right)
= 8.3604801843+-6.0742444084i \)