from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2448, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,12,8,3]))
pari: [g,chi] = znchar(Mod(1397,2448))
Basic properties
Modulus: | \(2448\) | |
Conductor: | \(2448\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2448.fa
\(\chi_{2448}(5,\cdot)\) \(\chi_{2448}(317,\cdot)\) \(\chi_{2448}(437,\cdot)\) \(\chi_{2448}(581,\cdot)\) \(\chi_{2448}(605,\cdot)\) \(\chi_{2448}(653,\cdot)\) \(\chi_{2448}(677,\cdot)\) \(\chi_{2448}(821,\cdot)\) \(\chi_{2448}(941,\cdot)\) \(\chi_{2448}(1253,\cdot)\) \(\chi_{2448}(1397,\cdot)\) \(\chi_{2448}(1469,\cdot)\) \(\chi_{2448}(1757,\cdot)\) \(\chi_{2448}(1949,\cdot)\) \(\chi_{2448}(2237,\cdot)\) \(\chi_{2448}(2309,\cdot)\)
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
\((2143,613,1361,1873)\) → \((1,i,e\left(\frac{1}{6}\right),e\left(\frac{1}{16}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 2448 }(1397, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)