sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(65, base_ring=CyclotomicField(6))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,2]))
pari: [g,chi] = znchar(Mod(16,65))
Basic properties
| Modulus: | \(65\) | |
| Conductor: | \(13\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(3\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{13}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 65.e
\(\chi_{65}(16,\cdot)\) \(\chi_{65}(61,\cdot)\)
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: | \(\Q(\sqrt{-3}) \) |
| Fixed field: | 3.3.169.1 |
Values on generators
\((27,41)\) → \((1,e\left(\frac{1}{3}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(1\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{65}(16,\cdot)) = \sum_{r\in \Z/65\Z} \chi_{65}(16,r) e\left(\frac{2r}{65}\right) = -2.565804281+2.5331104184i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{65}(16,\cdot),\chi_{65}(1,\cdot)) = \sum_{r\in \Z/65\Z} \chi_{65}(16,r) \chi_{65}(1,1-r) = -3 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{65}(16,·))
= \sum_{r \in \Z/65\Z}
\chi_{65}(16,r) e\left(\frac{1 r + 2 r^{-1}}{65}\right)
= 1.4986674202+2.5957681155i \)