sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(259, base_ring=CyclotomicField(18))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([6,4]))
pari: [g,chi] = znchar(Mod(219,259))
Basic properties
| Modulus: | \(259\) | |
| Conductor: | \(259\) | 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 259.x
\(\chi_{259}(44,\cdot)\) \(\chi_{259}(46,\cdot)\) \(\chi_{259}(53,\cdot)\) \(\chi_{259}(107,\cdot)\) \(\chi_{259}(123,\cdot)\) \(\chi_{259}(219,\cdot)\)
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
\((38,113)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{2}{9}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{8}{9}\right)\) |
Related number fields
| Field of values: | \(\Q(\zeta_{9})\) |
| Fixed field: | 9.9.413239695274351729.2 |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{259}(219,\cdot)) = \sum_{r\in \Z/259\Z} \chi_{259}(219,r) e\left(\frac{2r}{259}\right) = 14.8591936239+-6.1809679539i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{259}(219,\cdot),\chi_{259}(1,\cdot)) = \sum_{r\in \Z/259\Z} \chi_{259}(219,r) \chi_{259}(1,1-r) = 1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{259}(219,·))
= \sum_{r \in \Z/259\Z}
\chi_{259}(219,r) e\left(\frac{1 r + 2 r^{-1}}{259}\right)
= -3.8007395579+1.3833560673i \)