sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(54)
sage: chi = H[31]
pari: [g,chi] = znchar(Mod(31,54))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 27 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 9 |
| 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 | = | Even |
| Orbit label | = | 54.e |
| Orbit index | = | 5 |
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 ]
\(\chi_{54}(7,\cdot)\) \(\chi_{54}(13,\cdot)\) \(\chi_{54}(25,\cdot)\) \(\chi_{54}(31,\cdot)\) \(\chi_{54}(43,\cdot)\) \(\chi_{54}(49,\cdot)\)
Inducing primitive character
Values on generators
\(29\) → \(e\left(\frac{1}{9}\right)\)
Values
| -1 | 1 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 25 | 29 | 31 |
| \(1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) |
Related number fields
| Field of values | \(\Q(\zeta_{9})\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{54}(31,\cdot)) = \sum_{r\in \Z/54\Z} \chi_{54}(31,r) e\left(\frac{r}{27}\right) = 3.7795443099+3.5658161492i \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{54}(31,\cdot),\chi_{54}(1,\cdot)) = \sum_{r\in \Z/54\Z} \chi_{54}(31,r) \chi_{54}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{54}(31,·))
= \sum_{r \in \Z/54\Z}
\chi_{54}(31,r) e\left(\frac{1 r + 2 r^{-1}}{54}\right)
= 8.4572335871+3.0781812899i \)