sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(45)
sage: chi = H[32]
pari: [g,chi] = znchar(Mod(32,45))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 45 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 12 |
| 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 | = | 45.l |
| Orbit index | = | 12 |
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 ]
\(\chi_{45}(2,\cdot)\) \(\chi_{45}(23,\cdot)\) \(\chi_{45}(32,\cdot)\) \(\chi_{45}(38,\cdot)\)
Values on generators
\((11,37)\) → \((e\left(\frac{5}{6}\right),i)\)
Values
| -1 | 1 | 2 | 4 | 7 | 8 | 11 | 13 | 14 | 16 | 17 | 19 |
| \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(i\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-i\) | \(-1\) |
Related number fields
| Field of values | \(\Q(\zeta_{12})\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{45}(32,\cdot)) = \sum_{r\in \Z/45\Z} \chi_{45}(32,r) e\left(\frac{2r}{45}\right) = 6.5684111193+1.362341869i \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{45}(32,\cdot),\chi_{45}(1,\cdot)) = \sum_{r\in \Z/45\Z} \chi_{45}(32,r) \chi_{45}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{45}(32,·))
= \sum_{r \in \Z/45\Z}
\chi_{45}(32,r) e\left(\frac{1 r + 2 r^{-1}}{45}\right)
= 4.8175775196+1.2908660059i \)