sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(64)
sage: chi = H[27]
pari: [g,chi] = znchar(Mod(27,64))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 64 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 16 |
| 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 | = | 64.j |
| Orbit index | = | 10 |
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 ]
\(\chi_{64}(3,\cdot)\) \(\chi_{64}(11,\cdot)\) \(\chi_{64}(19,\cdot)\) \(\chi_{64}(27,\cdot)\) \(\chi_{64}(35,\cdot)\) \(\chi_{64}(43,\cdot)\) \(\chi_{64}(51,\cdot)\) \(\chi_{64}(59,\cdot)\)
Values on generators
\((63,5)\) → \((-1,e\left(\frac{9}{16}\right))\)
Values
| -1 | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 |
| \(-1\) | \(1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(-i\) | \(-i\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) |
Related number fields
| Field of values | \(\Q(\zeta_{16})\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{64}(27,\cdot)) = \sum_{r\in \Z/64\Z} \chi_{64}(27,r) e\left(\frac{r}{32}\right) = 0.0 \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{64}(27,\cdot),\chi_{64}(1,\cdot)) = \sum_{r\in \Z/64\Z} \chi_{64}(27,r) \chi_{64}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{64}(27,·))
= \sum_{r \in \Z/64\Z}
\chi_{64}(27,r) e\left(\frac{1 r + 2 r^{-1}}{64}\right)
= -7.0553701148+-3.7711738946i \)