sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(189, base_ring=CyclotomicField(18))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([8,9]))
pari: [g,chi] = znchar(Mod(13,189))
Basic properties
| Modulus: | \(189\) | |
| Conductor: | \(189\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(18\) | 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 189.y
\(\chi_{189}(13,\cdot)\) \(\chi_{189}(34,\cdot)\) \(\chi_{189}(76,\cdot)\) \(\chi_{189}(97,\cdot)\) \(\chi_{189}(139,\cdot)\) \(\chi_{189}(160,\cdot)\)
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: | \(\Q(\zeta_{9})\) |
| Fixed field: | 18.0.39739057971752889532465351767.1 |
Values on generators
\((29,136)\) → \((e\left(\frac{4}{9}\right),-1)\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \(-1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{189}(13,\cdot)) = \sum_{r\in \Z/189\Z} \chi_{189}(13,r) e\left(\frac{2r}{189}\right) = -13.3771553221+-3.1704440524i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{189}(13,\cdot),\chi_{189}(1,\cdot)) = \sum_{r\in \Z/189\Z} \chi_{189}(13,r) \chi_{189}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{189}(13,·))
= \sum_{r \in \Z/189\Z}
\chi_{189}(13,r) e\left(\frac{1 r + 2 r^{-1}}{189}\right)
= 0.0 \)