sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(81)
sage: chi = H[29]
sage: H = DirichletGroup_conrey(81)
sage: chi = H[29]
pari: [g,chi] = znchar(Mod(29,81))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 81 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 54 |
| 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 | = | 81.h |
| Orbit index | = | 8 |
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 ]
\(\chi_{81}(2,\cdot)\) \(\chi_{81}(5,\cdot)\) \(\chi_{81}(11,\cdot)\) \(\chi_{81}(14,\cdot)\) \(\chi_{81}(20,\cdot)\) \(\chi_{81}(23,\cdot)\) \(\chi_{81}(29,\cdot)\) \(\chi_{81}(32,\cdot)\) \(\chi_{81}(38,\cdot)\) \(\chi_{81}(41,\cdot)\) \(\chi_{81}(47,\cdot)\) \(\chi_{81}(50,\cdot)\) \(\chi_{81}(56,\cdot)\) \(\chi_{81}(59,\cdot)\) \(\chi_{81}(65,\cdot)\) \(\chi_{81}(68,\cdot)\) \(\chi_{81}(74,\cdot)\) \(\chi_{81}(77,\cdot)\)
Values on generators
\(2\) → \(e\left(\frac{37}{54}\right)\)
Values
| -1 | 1 | 2 | 4 | 5 | 7 | 8 | 10 | 11 | 13 | 14 | 16 |
| \(-1\) | \(1\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{20}{27}\right)\) |
Related number fields
| Field of values | \(\Q(\zeta_{27})\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{81}(29,\cdot)) = \sum_{r\in \Z/81\Z} \chi_{81}(29,r) e\left(\frac{2r}{81}\right) = 5.9176923171+6.7809230669i \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{81}(29,\cdot),\chi_{81}(1,\cdot)) = \sum_{r\in \Z/81\Z} \chi_{81}(29,r) \chi_{81}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{81}(29,·))
= \sum_{r \in \Z/81\Z}
\chi_{81}(29,r) e\left(\frac{1 r + 2 r^{-1}}{81}\right)
= -0.0 \)