from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2944, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,19,16]))
pari: [g,chi] = znchar(Mod(2621,2944))
Basic properties
| Modulus: | \(2944\) | |
| Conductor: | \(2944\) | 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 2944.bf
\(\chi_{2944}(45,\cdot)\) \(\chi_{2944}(229,\cdot)\) \(\chi_{2944}(413,\cdot)\) \(\chi_{2944}(597,\cdot)\) \(\chi_{2944}(781,\cdot)\) \(\chi_{2944}(965,\cdot)\) \(\chi_{2944}(1149,\cdot)\) \(\chi_{2944}(1333,\cdot)\) \(\chi_{2944}(1517,\cdot)\) \(\chi_{2944}(1701,\cdot)\) \(\chi_{2944}(1885,\cdot)\) \(\chi_{2944}(2069,\cdot)\) \(\chi_{2944}(2253,\cdot)\) \(\chi_{2944}(2437,\cdot)\) \(\chi_{2944}(2621,\cdot)\) \(\chi_{2944}(2805,\cdot)\)
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: | 32.0.19247509741360815152884297845798278938249171496527684529937635790938725865750528.1 |
Values on generators
\((1151,645,2305)\) → \((1,e\left(\frac{19}{32}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2944 }(2621, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) |
sage: chi.jacobi_sum(n)