from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(203, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([7,27]))
pari: [g,chi] = znchar(Mod(129,203))
Basic properties
| Modulus: | \(203\) | |
| Conductor: | \(203\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | 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 203.u
\(\chi_{203}(5,\cdot)\) \(\chi_{203}(33,\cdot)\) \(\chi_{203}(38,\cdot)\) \(\chi_{203}(80,\cdot)\) \(\chi_{203}(96,\cdot)\) \(\chi_{203}(122,\cdot)\) \(\chi_{203}(129,\cdot)\) \(\chi_{203}(138,\cdot)\) \(\chi_{203}(150,\cdot)\) \(\chi_{203}(178,\cdot)\) \(\chi_{203}(180,\cdot)\) \(\chi_{203}(187,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{21})\) |
| Fixed field: | 42.0.409216671487706896433400972530400813921910324685198934110144125610159603082437813389667.1 |
Values on generators
\((59,176)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{9}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 203 }(129, a) \) | \(-1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)