sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(407, base_ring=CyclotomicField(18))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,8]))
pari: [g,chi] = znchar(Mod(342,407))
Basic properties
| Modulus: | \(407\) | |
| Conductor: | \(37\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(9\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{37}(9,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 407.l
\(\chi_{407}(12,\cdot)\) \(\chi_{407}(34,\cdot)\) \(\chi_{407}(144,\cdot)\) \(\chi_{407}(155,\cdot)\) \(\chi_{407}(342,\cdot)\) \(\chi_{407}(386,\cdot)\)
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: | 9.9.3512479453921.1 |
Values on generators
\((112,298)\) → \((1,e\left(\frac{4}{9}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \(1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{9}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{407}(342,\cdot)) = \sum_{r\in \Z/407\Z} \chi_{407}(342,r) e\left(\frac{2r}{407}\right) = -5.1558800098+-3.2275224747i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{407}(342,\cdot),\chi_{407}(1,\cdot)) = \sum_{r\in \Z/407\Z} \chi_{407}(342,r) \chi_{407}(1,1-r) = -9 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{407}(342,·))
= \sum_{r \in \Z/407\Z}
\chi_{407}(342,r) e\left(\frac{1 r + 2 r^{-1}}{407}\right)
= -5.3160485162+-30.1488093019i \)