sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(77)
sage: chi = H[51]
sage: H = DirichletGroup_conrey(77)
sage: chi = H[51]
pari: [g,chi] = znchar(Mod(51,77))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 77 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 30 |
| 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 | = | 77.o |
| Orbit index | = | 15 |
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 ]
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\(\chi_{77}(2,\cdot)\) \(\chi_{77}(18,\cdot)\) \(\chi_{77}(30,\cdot)\) \(\chi_{77}(39,\cdot)\) \(\chi_{77}(46,\cdot)\) \(\chi_{77}(51,\cdot)\) \(\chi_{77}(72,\cdot)\) \(\chi_{77}(74,\cdot)\)
Values on generators
\((45,57)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{7}{10}\right))\)
Values
| -1 | 1 | 2 | 3 | 4 | 5 | 6 | 8 | 9 | 10 | 12 | 13 |
| \(-1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{10}\right)\) |
Related number fields
| Field of values | \(\Q(\zeta_{15})\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{77}(51,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(51,r) e\left(\frac{2r}{77}\right) = -3.6183988583+-7.9941972519i \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{77}(51,\cdot),\chi_{77}(1,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(51,r) \chi_{77}(1,1-r) = 1 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{77}(51,·))
= \sum_{r \in \Z/77\Z}
\chi_{77}(51,r) e\left(\frac{1 r + 2 r^{-1}}{77}\right)
= -2.5146560404+1.1195970029i \)