sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(171, base_ring=CyclotomicField(18))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([6,2]))
pari: [g,chi] = znchar(Mod(4,171))
Basic properties
| Modulus: | \(171\) | |
| Conductor: | \(171\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(9\) | 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 171.w
\(\chi_{171}(4,\cdot)\) \(\chi_{171}(16,\cdot)\) \(\chi_{171}(43,\cdot)\) \(\chi_{171}(85,\cdot)\) \(\chi_{171}(139,\cdot)\) \(\chi_{171}(169,\cdot)\)
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
\((20,154)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{1}{9}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \(1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) |
Related number fields
| Field of values: | \(\Q(\zeta_{9})\) |
| Fixed field: | 9.9.9025761726072081.2 |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{171}(4,\cdot)) = \sum_{r\in \Z/171\Z} \chi_{171}(4,r) e\left(\frac{2r}{171}\right) = -4.9478795958+-12.1044821246i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{171}(4,\cdot),\chi_{171}(1,\cdot)) = \sum_{r\in \Z/171\Z} \chi_{171}(4,r) \chi_{171}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{171}(4,·))
= \sum_{r \in \Z/171\Z}
\chi_{171}(4,r) e\left(\frac{1 r + 2 r^{-1}}{171}\right)
= 4.120838395+23.3704358712i \)