from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1785, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,0,8,27]))
pari: [g,chi] = znchar(Mod(31,1785))
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 \(\chi_{119}(31,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1785.fi
\(\chi_{1785}(31,\cdot)\) \(\chi_{1785}(61,\cdot)\) \(\chi_{1785}(241,\cdot)\) \(\chi_{1785}(346,\cdot)\) \(\chi_{1785}(481,\cdot)\) \(\chi_{1785}(556,\cdot)\) \(\chi_{1785}(691,\cdot)\) \(\chi_{1785}(796,\cdot)\) \(\chi_{1785}(976,\cdot)\) \(\chi_{1785}(1006,\cdot)\) \(\chi_{1785}(1081,\cdot)\) \(\chi_{1785}(1111,\cdot)\) \(\chi_{1785}(1321,\cdot)\) \(\chi_{1785}(1501,\cdot)\) \(\chi_{1785}(1711,\cdot)\) \(\chi_{1785}(1741,\cdot)\)
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
\((596,1072,766,1261)\) → \((1,1,e\left(\frac{1}{6}\right),e\left(\frac{9}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(19\) | \(22\) | \(23\) | \(26\) |
| \( \chi_{ 1785 }(31, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(-i\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) |
sage: chi.jacobi_sum(n)