sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(67)
sage: chi = H[50]
pari: [g,chi] = znchar(Mod(50,67))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 67 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 66 |
| 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 | = | 67.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 ]
\(\chi_{67}(2,\cdot)\) \(\chi_{67}(7,\cdot)\) \(\chi_{67}(11,\cdot)\) \(\chi_{67}(12,\cdot)\) \(\chi_{67}(13,\cdot)\) \(\chi_{67}(18,\cdot)\) \(\chi_{67}(20,\cdot)\) \(\chi_{67}(28,\cdot)\) \(\chi_{67}(31,\cdot)\) \(\chi_{67}(32,\cdot)\) \(\chi_{67}(34,\cdot)\) \(\chi_{67}(41,\cdot)\) \(\chi_{67}(44,\cdot)\) \(\chi_{67}(46,\cdot)\) \(\chi_{67}(48,\cdot)\) \(\chi_{67}(50,\cdot)\) \(\chi_{67}(51,\cdot)\) \(\chi_{67}(57,\cdot)\) \(\chi_{67}(61,\cdot)\) \(\chi_{67}(63,\cdot)\)
Values on generators
\(2\) → \(e\left(\frac{31}{66}\right)\)
Values
| -1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| \(-1\) | \(1\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{47}{66}\right)\) |
Related number fields
| Field of values | \(\Q(\zeta_{33})\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{67}(50,\cdot)) = \sum_{r\in \Z/67\Z} \chi_{67}(50,r) e\left(\frac{2r}{67}\right) = 2.7261195115+-7.7180484845i \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{67}(50,\cdot),\chi_{67}(1,\cdot)) = \sum_{r\in \Z/67\Z} \chi_{67}(50,r) \chi_{67}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{67}(50,·))
= \sum_{r \in \Z/67\Z}
\chi_{67}(50,r) e\left(\frac{1 r + 2 r^{-1}}{67}\right)
= -9.3959352816+0.8972030358i \)