from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2944, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,5,16]))
pari: [g,chi] = znchar(Mod(1333,2944))
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
\((1151,645,2305)\) → \((1,e\left(\frac{5}{32}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 2944 }(1333, a) \) | \(-1\) | \(1\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) |
sage: chi.jacobi_sum(n)