sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(67, base_ring=CyclotomicField(66))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([8]))
pari: [g,chi] = znchar(Mod(55,67))
Basic properties
| Modulus: | \(67\) | |
| Conductor: | \(67\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(33\) | 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 67.g
\(\chi_{67}(4,\cdot)\) \(\chi_{67}(6,\cdot)\) \(\chi_{67}(10,\cdot)\) \(\chi_{67}(16,\cdot)\) \(\chi_{67}(17,\cdot)\) \(\chi_{67}(19,\cdot)\) \(\chi_{67}(21,\cdot)\) \(\chi_{67}(23,\cdot)\) \(\chi_{67}(26,\cdot)\) \(\chi_{67}(33,\cdot)\) \(\chi_{67}(35,\cdot)\) \(\chi_{67}(36,\cdot)\) \(\chi_{67}(39,\cdot)\) \(\chi_{67}(47,\cdot)\) \(\chi_{67}(49,\cdot)\) \(\chi_{67}(54,\cdot)\) \(\chi_{67}(55,\cdot)\) \(\chi_{67}(56,\cdot)\) \(\chi_{67}(60,\cdot)\) \(\chi_{67}(65,\cdot)\)
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
\(2\) → \(e\left(\frac{4}{33}\right)\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \(1\) | \(1\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) |
Related number fields
| Field of values: | \(\Q(\zeta_{33})\) |
| Fixed field: | \(\Q(\zeta_{67})^+\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{67}(55,\cdot)) = \sum_{r\in \Z/67\Z} \chi_{67}(55,r) e\left(\frac{2r}{67}\right) = 5.2808267405+-6.2540282168i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{67}(55,\cdot),\chi_{67}(1,\cdot)) = \sum_{r\in \Z/67\Z} \chi_{67}(55,r) \chi_{67}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{67}(55,·))
= \sum_{r \in \Z/67\Z}
\chi_{67}(55,r) e\left(\frac{1 r + 2 r^{-1}}{67}\right)
= 11.5057236472+4.6061969105i \)