sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(87)
sage: chi = H[11]
sage: H = DirichletGroup_conrey(87)
sage: chi = H[11]
pari: [g,chi] = znchar(Mod(11,87))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 87 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 28 |
| 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 | = | 87.k |
| Orbit index | = | 11 |
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 ]
\(\chi_{87}(2,\cdot)\) \(\chi_{87}(8,\cdot)\) \(\chi_{87}(11,\cdot)\) \(\chi_{87}(14,\cdot)\) \(\chi_{87}(26,\cdot)\) \(\chi_{87}(32,\cdot)\) \(\chi_{87}(44,\cdot)\) \(\chi_{87}(47,\cdot)\) \(\chi_{87}(50,\cdot)\) \(\chi_{87}(56,\cdot)\) \(\chi_{87}(68,\cdot)\) \(\chi_{87}(77,\cdot)\)
Values on generators
\((59,31)\) → \((-1,e\left(\frac{25}{28}\right))\)
Values
| -1 | 1 | 2 | 4 | 5 | 7 | 8 | 10 | 11 | 13 | 14 | 16 |
| \(1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) |
Related number fields
| Field of values | \(\Q(\zeta_{28})\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{87}(11,\cdot)) = \sum_{r\in \Z/87\Z} \chi_{87}(11,r) e\left(\frac{2r}{87}\right) = -2.3611785129+-9.0235711351i \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{87}(11,\cdot),\chi_{87}(1,\cdot)) = \sum_{r\in \Z/87\Z} \chi_{87}(11,r) \chi_{87}(1,1-r) = 1 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{87}(11,·))
= \sum_{r \in \Z/87\Z}
\chi_{87}(11,r) e\left(\frac{1 r + 2 r^{-1}}{87}\right)
= 0.0 \)