sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(47)
sage: chi = H[12]
pari: [g,chi] = znchar(Mod(12,47))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 47 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 23 |
| 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 | = | 47.c |
| Orbit index | = | 3 |
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 ]
\(\chi_{47}(2,\cdot)\) \(\chi_{47}(3,\cdot)\) \(\chi_{47}(4,\cdot)\) \(\chi_{47}(6,\cdot)\) \(\chi_{47}(7,\cdot)\) \(\chi_{47}(8,\cdot)\) \(\chi_{47}(9,\cdot)\) \(\chi_{47}(12,\cdot)\) \(\chi_{47}(14,\cdot)\) \(\chi_{47}(16,\cdot)\) \(\chi_{47}(17,\cdot)\) \(\chi_{47}(18,\cdot)\) \(\chi_{47}(21,\cdot)\) \(\chi_{47}(24,\cdot)\) \(\chi_{47}(25,\cdot)\) \(\chi_{47}(27,\cdot)\) \(\chi_{47}(28,\cdot)\) \(\chi_{47}(32,\cdot)\) \(\chi_{47}(34,\cdot)\) \(\chi_{47}(36,\cdot)\) \(\chi_{47}(37,\cdot)\) \(\chi_{47}(42,\cdot)\)
Values on generators
\(5\) → \(e\left(\frac{5}{23}\right)\)
Values
| -1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| \(1\) | \(1\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) |
Related number fields
| Field of values | \(\Q(\zeta_{23})\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{47}(12,\cdot)) = \sum_{r\in \Z/47\Z} \chi_{47}(12,r) e\left(\frac{2r}{47}\right) = 5.8952695944+3.4993994355i \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{47}(12,\cdot),\chi_{47}(1,\cdot)) = \sum_{r\in \Z/47\Z} \chi_{47}(12,r) \chi_{47}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{47}(12,·))
= \sum_{r \in \Z/47\Z}
\chi_{47}(12,r) e\left(\frac{1 r + 2 r^{-1}}{47}\right)
= 1.5512253922+-0.4346329717i \)