from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4160, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,9,6,2]))
pari: [g,chi] = znchar(Mod(457,4160))
Basic properties
| Modulus: | \(4160\) | |
| Conductor: | \(2080\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2080}(1757,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4160.iq
\(\chi_{4160}(457,\cdot)\) \(\chi_{4160}(617,\cdot)\) \(\chi_{4160}(1593,\cdot)\) \(\chi_{4160}(1753,\cdot)\) \(\chi_{4160}(2537,\cdot)\) \(\chi_{4160}(2697,\cdot)\) \(\chi_{4160}(3673,\cdot)\) \(\chi_{4160}(3833,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
| Fixed field: | Number field defined by a degree 24 polynomial |
Values on generators
\((4031,261,2497,1601)\) → \((1,e\left(\frac{3}{8}\right),i,e\left(\frac{1}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 4160 }(457, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{23}{24}\right)\) |
sage: chi.jacobi_sum(n)