sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(191, base_ring=CyclotomicField(38))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([20]))
pari: [g,chi] = znchar(Mod(25,191))
Basic properties
| Modulus: | \(191\) | |
| Conductor: | \(191\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(19\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 191.e
\(\chi_{191}(5,\cdot)\) \(\chi_{191}(6,\cdot)\) \(\chi_{191}(25,\cdot)\) \(\chi_{191}(30,\cdot)\) \(\chi_{191}(32,\cdot)\) \(\chi_{191}(36,\cdot)\) \(\chi_{191}(52,\cdot)\) \(\chi_{191}(69,\cdot)\) \(\chi_{191}(107,\cdot)\) \(\chi_{191}(121,\cdot)\) \(\chi_{191}(125,\cdot)\) \(\chi_{191}(136,\cdot)\) \(\chi_{191}(150,\cdot)\) \(\chi_{191}(153,\cdot)\) \(\chi_{191}(154,\cdot)\) \(\chi_{191}(160,\cdot)\) \(\chi_{191}(177,\cdot)\) \(\chi_{191}(180,\cdot)\)
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{19})\) |
| Fixed field: | 19.19.114445997944945591651333831028437092270721.1 |
Values on generators
\(19\) → \(e\left(\frac{10}{19}\right)\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \(1\) | \(1\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(1\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{191}(25,\cdot)) = \sum_{r\in \Z/191\Z} \chi_{191}(25,r) e\left(\frac{2r}{191}\right) = 7.1639507088+11.8185367217i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{191}(25,\cdot),\chi_{191}(1,\cdot)) = \sum_{r\in \Z/191\Z} \chi_{191}(25,r) \chi_{191}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{191}(25,·))
= \sum_{r \in \Z/191\Z}
\chi_{191}(25,r) e\left(\frac{1 r + 2 r^{-1}}{191}\right)
= -0.1075917144+-0.0582257241i \)