sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(191, base_ring=CyclotomicField(38))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([27]))
pari: [g,chi] = znchar(Mod(55,191))
Basic properties
| Modulus: | \(191\) | |
| Conductor: | \(191\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 191.f
\(\chi_{191}(11,\cdot)\) \(\chi_{191}(14,\cdot)\) \(\chi_{191}(31,\cdot)\) \(\chi_{191}(37,\cdot)\) \(\chi_{191}(38,\cdot)\) \(\chi_{191}(41,\cdot)\) \(\chi_{191}(55,\cdot)\) \(\chi_{191}(66,\cdot)\) \(\chi_{191}(70,\cdot)\) \(\chi_{191}(84,\cdot)\) \(\chi_{191}(122,\cdot)\) \(\chi_{191}(139,\cdot)\) \(\chi_{191}(155,\cdot)\) \(\chi_{191}(159,\cdot)\) \(\chi_{191}(161,\cdot)\) \(\chi_{191}(166,\cdot)\) \(\chi_{191}(185,\cdot)\) \(\chi_{191}(186,\cdot)\)
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Values on generators
\(19\) → \(e\left(\frac{27}{38}\right)\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \(-1\) | \(1\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(-1\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) |
Related number fields
| Field of values: | \(\Q(\zeta_{19})\) |
| Fixed field: | 38.0.2501696311112367702213593384284049957523786814936027691413131289153060507631187229631.1 |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{191}(55,\cdot)) = \sum_{r\in \Z/191\Z} \chi_{191}(55,r) e\left(\frac{2r}{191}\right) = 3.9575665655+13.2415130132i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{191}(55,\cdot),\chi_{191}(1,\cdot)) = \sum_{r\in \Z/191\Z} \chi_{191}(55,r) \chi_{191}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{191}(55,·))
= \sum_{r \in \Z/191\Z}
\chi_{191}(55,r) e\left(\frac{1 r + 2 r^{-1}}{191}\right)
= -10.4225400069+9.5946239803i \)