from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1785, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,0,32,39]))
pari: [g,chi] = znchar(Mod(46,1785))
Basic properties
| Modulus: | \(1785\) | |
| Conductor: | \(119\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{119}(46,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1785.fe
\(\chi_{1785}(46,\cdot)\) \(\chi_{1785}(226,\cdot)\) \(\chi_{1785}(436,\cdot)\) \(\chi_{1785}(466,\cdot)\) \(\chi_{1785}(541,\cdot)\) \(\chi_{1785}(571,\cdot)\) \(\chi_{1785}(751,\cdot)\) \(\chi_{1785}(856,\cdot)\) \(\chi_{1785}(991,\cdot)\) \(\chi_{1785}(1066,\cdot)\) \(\chi_{1785}(1201,\cdot)\) \(\chi_{1785}(1306,\cdot)\) \(\chi_{1785}(1486,\cdot)\) \(\chi_{1785}(1516,\cdot)\) \(\chi_{1785}(1591,\cdot)\) \(\chi_{1785}(1621,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((596,1072,766,1261)\) → \((1,1,e\left(\frac{2}{3}\right),e\left(\frac{13}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(19\) | \(22\) | \(23\) | \(26\) |
| \( \chi_{ 1785 }(46, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(i\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) |
sage: chi.jacobi_sum(n)