sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(145, base_ring=CyclotomicField(28))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([14,5]))
pari: [g,chi] = znchar(Mod(119,145))
Basic properties
| Modulus: | \(145\) | |
| Conductor: | \(145\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | 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 145.s
\(\chi_{145}(14,\cdot)\) \(\chi_{145}(19,\cdot)\) \(\chi_{145}(39,\cdot)\) \(\chi_{145}(44,\cdot)\) \(\chi_{145}(69,\cdot)\) \(\chi_{145}(79,\cdot)\) \(\chi_{145}(84,\cdot)\) \(\chi_{145}(89,\cdot)\) \(\chi_{145}(114,\cdot)\) \(\chi_{145}(119,\cdot)\) \(\chi_{145}(124,\cdot)\) \(\chi_{145}(134,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
| Fixed field: | 28.0.18634854406558377293367533932433545897882080078125.1 |
Values on generators
\((117,31)\) → \((-1,e\left(\frac{5}{28}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \(-1\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(-i\) | \(e\left(\frac{5}{7}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{145}(119,\cdot)) = \sum_{r\in \Z/145\Z} \chi_{145}(119,r) e\left(\frac{2r}{145}\right) = -8.6189080403+-8.4091868925i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{145}(119,\cdot),\chi_{145}(1,\cdot)) = \sum_{r\in \Z/145\Z} \chi_{145}(119,r) \chi_{145}(1,1-r) = 1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{145}(119,·))
= \sum_{r \in \Z/145\Z}
\chi_{145}(119,r) e\left(\frac{1 r + 2 r^{-1}}{145}\right)
= -0.0 \)