sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(113)
sage: chi = H[11]
pari: [g,chi] = znchar(Mod(11,113))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 113 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 56 |
| 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 | = | Even |
| Orbit label | = | 113.i |
| Orbit index | = | 9 |
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 ]
\(\chi_{113}(9,\cdot)\) \(\chi_{113}(11,\cdot)\) \(\chi_{113}(13,\cdot)\) \(\chi_{113}(22,\cdot)\) \(\chi_{113}(25,\cdot)\) \(\chi_{113}(26,\cdot)\) \(\chi_{113}(31,\cdot)\) \(\chi_{113}(36,\cdot)\) \(\chi_{113}(41,\cdot)\) \(\chi_{113}(50,\cdot)\) \(\chi_{113}(51,\cdot)\) \(\chi_{113}(52,\cdot)\) \(\chi_{113}(61,\cdot)\) \(\chi_{113}(62,\cdot)\) \(\chi_{113}(63,\cdot)\) \(\chi_{113}(72,\cdot)\) \(\chi_{113}(77,\cdot)\) \(\chi_{113}(82,\cdot)\) \(\chi_{113}(87,\cdot)\) \(\chi_{113}(88,\cdot)\) \(\chi_{113}(91,\cdot)\) \(\chi_{113}(100,\cdot)\) \(\chi_{113}(102,\cdot)\) \(\chi_{113}(104,\cdot)\)
Values on generators
\(3\) → \(e\left(\frac{37}{56}\right)\)
Values
| -1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| \(1\) | \(1\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{25}{28}\right)\) |
Related number fields
| Field of values | \(\Q(\zeta_{56})\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{113}(11,\cdot)) = \sum_{r\in \Z/113\Z} \chi_{113}(11,r) e\left(\frac{2r}{113}\right) = 2.4143389424+-10.3523411589i \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{113}(11,\cdot),\chi_{113}(1,\cdot)) = \sum_{r\in \Z/113\Z} \chi_{113}(11,r) \chi_{113}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{113}(11,·))
= \sum_{r \in \Z/113\Z}
\chi_{113}(11,r) e\left(\frac{1 r + 2 r^{-1}}{113}\right)
= 6.782139535+-1.5479790913i \)