from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2368, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,13,12]))
pari: [g,chi] = znchar(Mod(43,2368))
Basic properties
| Modulus: | \(2368\) | |
| Conductor: | \(2368\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | 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 2368.by
\(\chi_{2368}(43,\cdot)\) \(\chi_{2368}(179,\cdot)\) \(\chi_{2368}(635,\cdot)\) \(\chi_{2368}(771,\cdot)\) \(\chi_{2368}(1227,\cdot)\) \(\chi_{2368}(1363,\cdot)\) \(\chi_{2368}(1819,\cdot)\) \(\chi_{2368}(1955,\cdot)\)
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: | \(\Q(\zeta_{16})\) |
| Fixed field: | 16.16.3979150324571965896775823457783990324297728.1 |
Values on generators
\((1407,1925,705)\) → \((-1,e\left(\frac{13}{16}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2368 }(43, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) |
sage: chi.jacobi_sum(n)