from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4224, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,19,16,16]))
pari: [g,chi] = znchar(Mod(2243,4224))
Basic properties
| Modulus: | \(4224\) | |
| Conductor: | \(4224\) | 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 4224.cp
\(\chi_{4224}(131,\cdot)\) \(\chi_{4224}(395,\cdot)\) \(\chi_{4224}(659,\cdot)\) \(\chi_{4224}(923,\cdot)\) \(\chi_{4224}(1187,\cdot)\) \(\chi_{4224}(1451,\cdot)\) \(\chi_{4224}(1715,\cdot)\) \(\chi_{4224}(1979,\cdot)\) \(\chi_{4224}(2243,\cdot)\) \(\chi_{4224}(2507,\cdot)\) \(\chi_{4224}(2771,\cdot)\) \(\chi_{4224}(3035,\cdot)\) \(\chi_{4224}(3299,\cdot)\) \(\chi_{4224}(3563,\cdot)\) \(\chi_{4224}(3827,\cdot)\) \(\chi_{4224}(4091,\cdot)\)
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: | 32.0.6208007170334900849388551577113100621683540552916899074163477799595412799816204288.1 |
Values on generators
\((2047,133,1409,3841)\) → \((-1,e\left(\frac{19}{32}\right),-1,-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4224 }(2243, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(i\) | \(e\left(\frac{1}{32}\right)\) |
sage: chi.jacobi_sum(n)