from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2240, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,9,0,40]))
pari: [g,chi] = znchar(Mod(61,2240))
Basic properties
| Modulus: | \(2240\) | |
| Conductor: | \(448\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{448}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2240.fd
\(\chi_{2240}(61,\cdot)\) \(\chi_{2240}(101,\cdot)\) \(\chi_{2240}(341,\cdot)\) \(\chi_{2240}(381,\cdot)\) \(\chi_{2240}(621,\cdot)\) \(\chi_{2240}(661,\cdot)\) \(\chi_{2240}(901,\cdot)\) \(\chi_{2240}(941,\cdot)\) \(\chi_{2240}(1181,\cdot)\) \(\chi_{2240}(1221,\cdot)\) \(\chi_{2240}(1461,\cdot)\) \(\chi_{2240}(1501,\cdot)\) \(\chi_{2240}(1741,\cdot)\) \(\chi_{2240}(1781,\cdot)\) \(\chi_{2240}(2021,\cdot)\) \(\chi_{2240}(2061,\cdot)\)
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
\((1471,1541,897,1921)\) → \((1,e\left(\frac{3}{16}\right),1,e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 2240 }(61, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)