from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1444, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,14]))
pari: [g,chi] = znchar(Mod(39,1444))
Basic properties
| Modulus: | \(1444\) | |
| Conductor: | \(1444\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | 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 1444.p
\(\chi_{1444}(39,\cdot)\) \(\chi_{1444}(115,\cdot)\) \(\chi_{1444}(191,\cdot)\) \(\chi_{1444}(267,\cdot)\) \(\chi_{1444}(343,\cdot)\) \(\chi_{1444}(419,\cdot)\) \(\chi_{1444}(495,\cdot)\) \(\chi_{1444}(571,\cdot)\) \(\chi_{1444}(647,\cdot)\) \(\chi_{1444}(799,\cdot)\) \(\chi_{1444}(875,\cdot)\) \(\chi_{1444}(951,\cdot)\) \(\chi_{1444}(1027,\cdot)\) \(\chi_{1444}(1103,\cdot)\) \(\chi_{1444}(1179,\cdot)\) \(\chi_{1444}(1255,\cdot)\) \(\chi_{1444}(1331,\cdot)\) \(\chi_{1444}(1407,\cdot)\)
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: | 38.0.32314623763625504522847581826131926264699228491488831973421099549792171888378286207518194939116004573184.1 |
Values on generators
\((723,1085)\) → \((-1,e\left(\frac{7}{19}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 1444 }(39, a) \) | \(-1\) | \(1\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) |
sage: chi.jacobi_sum(n)