from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4224, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,19,0,0]))
pari: [g,chi] = znchar(Mod(67,4224))
Basic properties
| Modulus: | \(4224\) | |
| Conductor: | \(128\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{128}(67,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4224.cm
\(\chi_{4224}(67,\cdot)\) \(\chi_{4224}(331,\cdot)\) \(\chi_{4224}(595,\cdot)\) \(\chi_{4224}(859,\cdot)\) \(\chi_{4224}(1123,\cdot)\) \(\chi_{4224}(1387,\cdot)\) \(\chi_{4224}(1651,\cdot)\) \(\chi_{4224}(1915,\cdot)\) \(\chi_{4224}(2179,\cdot)\) \(\chi_{4224}(2443,\cdot)\) \(\chi_{4224}(2707,\cdot)\) \(\chi_{4224}(2971,\cdot)\) \(\chi_{4224}(3235,\cdot)\) \(\chi_{4224}(3499,\cdot)\) \(\chi_{4224}(3763,\cdot)\) \(\chi_{4224}(4027,\cdot)\)
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: | 32.0.3138550867693340381917894711603833208051177722232017256448.1 |
Values on generators
\((2047,133,1409,3841)\) → \((-1,e\left(\frac{19}{32}\right),1,1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4224 }(67, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(i\) | \(e\left(\frac{1}{32}\right)\) |
sage: chi.jacobi_sum(n)