from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2448, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,0,16,27]))
pari: [g,chi] = znchar(Mod(31,2448))
Basic properties
Modulus: | \(2448\) | |
Conductor: | \(612\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{612}(31,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2448.fd
\(\chi_{2448}(31,\cdot)\) \(\chi_{2448}(79,\cdot)\) \(\chi_{2448}(175,\cdot)\) \(\chi_{2448}(367,\cdot)\) \(\chi_{2448}(607,\cdot)\) \(\chi_{2448}(751,\cdot)\) \(\chi_{2448}(895,\cdot)\) \(\chi_{2448}(1183,\cdot)\) \(\chi_{2448}(1231,\cdot)\) \(\chi_{2448}(1519,\cdot)\) \(\chi_{2448}(1663,\cdot)\) \(\chi_{2448}(1807,\cdot)\) \(\chi_{2448}(2047,\cdot)\) \(\chi_{2448}(2239,\cdot)\) \(\chi_{2448}(2335,\cdot)\) \(\chi_{2448}(2383,\cdot)\)
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
\((2143,613,1361,1873)\) → \((-1,1,e\left(\frac{1}{3}\right),e\left(\frac{9}{16}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 2448 }(31, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)