from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(722, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([14]))
pari: [g,chi] = znchar(Mod(39,722))
Basic properties
| Modulus: | \(722\) | |
| Conductor: | \(361\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(19\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{361}(39,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 722.g
\(\chi_{722}(39,\cdot)\) \(\chi_{722}(77,\cdot)\) \(\chi_{722}(115,\cdot)\) \(\chi_{722}(153,\cdot)\) \(\chi_{722}(191,\cdot)\) \(\chi_{722}(229,\cdot)\) \(\chi_{722}(267,\cdot)\) \(\chi_{722}(305,\cdot)\) \(\chi_{722}(343,\cdot)\) \(\chi_{722}(381,\cdot)\) \(\chi_{722}(419,\cdot)\) \(\chi_{722}(457,\cdot)\) \(\chi_{722}(495,\cdot)\) \(\chi_{722}(533,\cdot)\) \(\chi_{722}(571,\cdot)\) \(\chi_{722}(609,\cdot)\) \(\chi_{722}(647,\cdot)\) \(\chi_{722}(685,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
| Fixed field: | 19.19.10842505080063916320800450434338728415281531281.1 |
Values on generators
\(363\) → \(e\left(\frac{7}{19}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 722 }(39, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) |
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)