from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2176, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,17,16]))
pari: [g,chi] = znchar(Mod(2107,2176))
Basic properties
| Modulus: | \(2176\) | |
| Conductor: | \(2176\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2176.dz
\(\chi_{2176}(67,\cdot)\) \(\chi_{2176}(203,\cdot)\) \(\chi_{2176}(339,\cdot)\) \(\chi_{2176}(475,\cdot)\) \(\chi_{2176}(611,\cdot)\) \(\chi_{2176}(747,\cdot)\) \(\chi_{2176}(883,\cdot)\) \(\chi_{2176}(1019,\cdot)\) \(\chi_{2176}(1155,\cdot)\) \(\chi_{2176}(1291,\cdot)\) \(\chi_{2176}(1427,\cdot)\) \(\chi_{2176}(1563,\cdot)\) \(\chi_{2176}(1699,\cdot)\) \(\chi_{2176}(1835,\cdot)\) \(\chi_{2176}(1971,\cdot)\) \(\chi_{2176}(2107,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
| Fixed field: | 32.0.152725625984366375633872344931707463035520335286488090842213114237373265215488.1 |
Values on generators
\((511,1157,513)\) → \((-1,e\left(\frac{17}{32}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 2176 }(2107, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) |
sage: chi.jacobi_sum(n)