from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2240, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,9,36,8]))
pari: [g,chi] = znchar(Mod(3,2240))
Basic properties
| Modulus: | \(2240\) | |
| Conductor: | \(2240\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | 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 2240.fm
\(\chi_{2240}(3,\cdot)\) \(\chi_{2240}(187,\cdot)\) \(\chi_{2240}(243,\cdot)\) \(\chi_{2240}(507,\cdot)\) \(\chi_{2240}(563,\cdot)\) \(\chi_{2240}(747,\cdot)\) \(\chi_{2240}(803,\cdot)\) \(\chi_{2240}(1067,\cdot)\) \(\chi_{2240}(1123,\cdot)\) \(\chi_{2240}(1307,\cdot)\) \(\chi_{2240}(1363,\cdot)\) \(\chi_{2240}(1627,\cdot)\) \(\chi_{2240}(1683,\cdot)\) \(\chi_{2240}(1867,\cdot)\) \(\chi_{2240}(1923,\cdot)\) \(\chi_{2240}(2187,\cdot)\)
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
\((1471,1541,897,1921)\) → \((-1,e\left(\frac{3}{16}\right),-i,e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 2240 }(3, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)