sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(51)
sage: chi = H[40]
pari: [g,chi] = znchar(Mod(40,51))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 17 |
| 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 | = | No |
| sage: chi.is_odd()
pari: zncharisodd(g,chi)
| ||
| Parity | = | Odd |
| Orbit label | = | 51.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 ]
\(\chi_{51}(7,\cdot)\) \(\chi_{51}(10,\cdot)\) \(\chi_{51}(22,\cdot)\) \(\chi_{51}(28,\cdot)\) \(\chi_{51}(31,\cdot)\) \(\chi_{51}(37,\cdot)\) \(\chi_{51}(40,\cdot)\) \(\chi_{51}(46,\cdot)\)
Inducing primitive character
Values on generators
\((35,37)\) → \((1,e\left(\frac{15}{16}\right))\)
Values
| -1 | 1 | 2 | 4 | 5 | 7 | 8 | 10 | 11 | 13 | 14 | 16 |
| \(-1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(-i\) | \(e\left(\frac{7}{16}\right)\) | \(-1\) |
Related number fields
| Field of values | \(\Q(\zeta_{16})\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{51}(40,\cdot)) = \sum_{r\in \Z/51\Z} \chi_{51}(40,r) e\left(\frac{2r}{51}\right) = -2.2558909682+-3.4512252809i \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{51}(40,\cdot),\chi_{51}(1,\cdot)) = \sum_{r\in \Z/51\Z} \chi_{51}(40,r) \chi_{51}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{51}(40,·))
= \sum_{r \in \Z/51\Z}
\chi_{51}(40,r) e\left(\frac{1 r + 2 r^{-1}}{51}\right)
= -4.0113411222+9.6842341405i \)